In just six short lines, "Chef" seems to describe the strange mix of pride and loneliness that can come with skilled work. The speaker talks about making "our work last" in the kitchen, and calls the effort "so handsome a trouble." That phrase is a nice contradiction: the job is hard and troublesome, but also beautiful and worth doing. The word "our" hints that the work is shared, perhaps with a team, though the poem doesn't say.
The ending shifts the tone. The speaker compares himself to "a prince come to court," someone who arrives with status and expectation, only to be "welcome to nobody." Judging by the imagery, the feeling is a quiet sting of being unappreciated. Food is made to vanish, so the effort rarely gets noticed. The mood is wry, a little weary, and tender about the craft. The speaker takes pride in the work but admits it often goes unrecognized.