This is a literal phrase taken from one of John's letters to Abigail

Mr. Hemenway came and kindly invited us to dine, but we had engaged a Dinner at Littlefields, so we returned there, dined and took our Horses to Meeting in the Afternoon...

In this context it seems he means "engaged" to be "accepted". But I mostly wanted to point out that this is one man's phrase from almost 250 years ago, it's hard to say if it was a regional phrase or his personal lingo he used with his wife or a typo or what. I searched "engaged a breakfast", "engaged a lunch" and "engaged a dinner" and this letter I linked is the only legitimate hit on the phrase.

I believe McCullough is referencing this ornate phrase as a nod to the letter. It's an Easter egg.

This is a literal phrase taken from one of John's letters to Abigail

Mr. Hemenway came and kindly invited us to dine, but we had engaged a Dinner at Littlefields, so we returned there, dined and took our Horses to Meeting in the Afternoon...

In this context it seems he means "engaged" to be "accepted". But I mostly wanted to point out that this is one man's phrase from almost 250 years ago, it's hard to say if it was a regional phrase or his personal lingo he used with his wife or a typo or what. I searched "engaged a breakfast", "engaged a lunch," "engaged a dinner", and "engaged a meal" and this letter I linked is the only legitimate hit on the phrase.

I believe McCullough is referencing this ornate phrase as a nod to the letter. It's an Easter egg.

EDIT: I did find this on oxforddictionaries.com, definition 3.2. Seems relevant to the letter quote, but not so much to yours.

dated Reserve (accommodation, a place, etc.) in advance. ‘she had offered to engage a room in the house of the woman’