"Stoothing" is a (colloquial1) dialect for studding/battening/lathing and plastering.
The earliest usage of "stoothing" I could find in Google Books was in 1789, in A Survey of the Lakes of Cumberland, Westmorland, and Lancashire:
These are a few of the changes which have been introduced of late days into the custom of building houses in Cumberland and the neighbouring parts within a few years past; and though I am willing to believe that modern fashions may have given more elegance to buildings, yet I am far from thinking that they have provided better in general for excluding either the wintry winds, or the heats of Summer. This objection, however, to modern improvements, will not hold in the comparison of the old chimneys with those that are now tiled: the balk, the randle-tree, the black-hood, and the stoothing, have little to boast of, but their superior utility in drying winter provisions. A circumstance of diminished consequence in these times, on account of the more general resort to markets, and the augmentation of rural commerce.
From 1828 in The Dialect of Craven with a Copious Glossary:
STOOTH, To lath and plaster. A. S. stuthe, a stake.
STOOTHING, Lathing and plastering.
From 1845 in The Laws Relating to Buildings, Comprising the Metropolitan Buildings Act:
(pg. 435) QUARTERING. - For the substantive [see PARTITION]; and for the verb, STOOTHING.
(pg. 442) STOOTHING, unde STOOTHED. - In carpentry, 'quartering;', i.e. the laying of battens against a wall, for the reception of laths.
As mentioned by Colin Fine in the comments, there is a citation of stoothing in 1788, but as a provincialism (specifically, of East Yorkshire, i.e., "steathing"):
(pg. 311) The oo before k changes into u long; as book, buke; to look, to luke: before t, l, m, th, generally into ea long; as boots, beats; fool, feal; broom, bream; tooth, teath: before r, mostly into ee; as floor, fleer; door, deer.
(pg. 356) STEATHING; a lath and plaister partition.
(From The Rural Economy of Yorkshire, 1788)
From The English Dialect Dictionary:
(pg. 742) STEATHING, sb. Obs. n.Cy. Dur. Yks. Also in form steeathen w.Dur. A partition of lath and plaster. Cf. stooth(e.
(pg. 786) STOOTH(E, v. Sc. Nhb. Cum. Yks. Lan. Also in form stothe n.Yks. ne.Yks. [stūð]. To lath and plaster a wall; to make in a room a false wall of battens, laths, and plaster. Cf. steathing.
Ayr., Slk. (Jam.), Gall. (A.W.), Cum., n.Yks. ne.Yks. Houses thus built are said to be 'steeath'd and daub'd.' m.Yks., w.Yks., ne.Lan.
Hence Stoothing or Stoothen, sb. a divisional wall of lath and plaster ; the surface 'stoothed'; esp. the lath and plaster work under a ceiling.
[Stuthe, stipare, Cath. Angl. (1483).]
"Stoothe/stooth/stoothing" is likely of Scottish origin and is a variant of stud.
From a letter from Philip Webb to George Washington Jack (dated 10th February, 1902):
I recognise some of the Scottish-building terms: are you going to "stoothe" the walls? Also, I hope you will be able to use the Ballachulish slates;' and if so, they should be laid on roof boarding, and have tenter hooks thus sketch to keep the gales from lifting them, and the more so that your roofs are not, so I fancy, as steep in pitch as at the big house.*
There were different variants of "stud" that were used as early as the (late) 13th century. From the Yorkshire Historical Dictionary (a reference generously provided by Henry in the comments):
- A stud, that is a short piece of timber, particularly such as were used for the uprights in a timber-framed house.
 ‘27 stodes, 14 ft in length’, (Scarborough)
[1434-5] ‘for stothes … for the house’, (Selby)
 vj quercus … pro les stothes grangić et les screnes et ostiis, (Beverley)
 iiij plough heads iij stothes a molde borde balkes, (South Cave)
 severall deales and stoothes, (Selby)
 90 yards of stoothing for the petitions [sic for partitions] at 1s a yard, (West Riding)
- An obsolete spelling of stud, that is the ornamental studs on a belt or girdle.
Used as a verb and noun:  a gyrdell stothed with sylver, (Clint)
 Unto ... my doughter a ledder belt with syluer stothes, (Thirsk)
 my best girdle harnessed with a rede corse and xxxiiij stothes of siluer and gilte, (Adel)
So the earliest usage of "stoothing" is from 1739.
From A Dictionary of the Scottish Language, 1846:
To STOOTH, r. a. To lath and plaster a wall, Ettr. For. Ayrs.
STOOTHED, part. adj. Apparently studded. Balteus vel balteum, a sword-belt or stoothed belt," Despaut. Gram.
STOOTHIN, s. Lathing and plastering, Ettr. For. Ayrs. — A.S. stuthe, palus, a pale or stake. Teut. stutte, id. stutt-en, fulcire; Isl. studd-r, suffultus.
From the The Dictionary of Architecture, 1887:
(pg. 145) STOOTH, stothe, stoothe, dorestothe. A thin spar of wood used in making a "stoothing" or partition of lath and plaster between rooms; as 1499, in SURTERS SOCIETY, York Fabric Rolls, 8vo., Durham, 1859, p.355. DURHAM HOUSEHOLD BOOK. Howden Roil, 5-6 king Edward VI, cir. 1530. "Stoothed" is a term used in the north of England and south of Scotland for battening with plaster over, as "quartering"; also "stouthing and warping". STOITHES of Finchale Priory.
(pg. 156) STUD (Fr. poteau de remplissage). Somersetshire SPEARING PEICE; Uprights; INTERQUARTER ; QUARTERS in Lancashire. "Stedes" (or studds) to put in the walls, 1557; CAMDEN SOCIETY, Ludlow Papers, 4to., 1867, p. 79. The posts, or quarters, in timber partitions and placed about 11 or 12 ins. apart according to the length of the laths for plastering; the spaces between being left open. BATTENING. REED. STOOTHE. PRICK POST. STORY POST.
From The Century Dictionary and Cyclopedia, Volume 7, 1900:
stoothing (stö'thing), n. [<stooth + -ing, or a var. of studding, accom. to stooth] Studding; battening.
1Alternate link: https://encyclopedia2.thefreedictionary.com/stoothing