We often reduce relative clauses when the relative pronoun is the subject of relative clause.
The professor who is teaching mathematics will leave the university. Reduced: The professor teaching mathematics will leave the university. (source)
"who is teaching mathematics" is the relative clause & the relative pronoun "who" is subject of relative clause.
But, I am not sure if the sentence "This is my last night working here" (Jennifer Aniston said in the Friend, at 0:40) can be interpreted as "This is my last night that I work here" & people removed "that I" & changed "work" to "working" or something like that???.
Also, see other example from the dictionary,
I will be busy cleaning the house. (source)
Why do they put a gerund right after an adjective like that?
Note: thoughtco says
The present participle is sometimes used as an adverb to describe the manner in which a verb is performed.
She taught pounding the grammar into their heads!
Angelo works considering all angles.
Notice how the present participle could be preceded with 'by' to give the same meaning:
She taught (by) pounding the grammar into their heads!
Angelo works (by) considering all angles.