Use “cooking” here, not “cuisine.”
“Cuisine-broil” is not a phrase. You have a hyphen where you should have a dash. Also, “broiled” ends in -ed, since it is a past participle (used as an adjective here).
“My mother brought the oyster soup” is good, but I’d say “My mother brought out the oyster soup.”
We should use “while” instead of “when,” to emphasize the on-going nature of the action. Actually, although your sentence is grammatically correct, I’d rephrase it to be more natural.
I added a comma after "fried cabbage." I do not believe in using the so-called Oxford comma all the time, but in this case, it makes the sentence a little clearer.
Here’s a possible rewrite, just be aware that there are many ways to say things, I tried to stay as close to your original as possible, since I can’t know exactly what you have in mind :)
"I took a look at the dishes on the table. Oh! They were my mother's best home cooking — broiled fish, fried cabbage, and lobster. I sat down and ate my favorite dishes with a feeling of great contentment. After I’d finished eating, my mother brought out the oyster soup."
By the way — I may stop by your house if I’m in the neighborhood.