Whenever I enter his house the first thing I notice (is/are) the big lamps.
Verb "to be" is connected to two key words: thing and lamps.
And this appears to be the problem. The sentence implies that "thing = the lamps"; but one is singular, whereas the other is plural. Both "is" and "are" seem to make sense, but at the same time it feels like they don't make any sense at all.
Let's remove unnecessary words of the sentence so that we are left with:
the thing (is/are) the lamps
If we put "is" there, it doesn't seem right, because of "the lamps is"
If we put "are" there, we get "the thing are"; again: gibberish
Which is correct? Is either correct?