I don't know the official title for this, but we used to order this appetizer from Arman's restaurant in Birmingham, AL all the time. That was back in '95, when the original Chef Arman had one of the most popular restaurants in the state.
It's still delicious and can actually be the entree for a nice summer meal. Tonight, we accompanied this one with toasted whole wheat almond bread with chevre and a bite of grilled chicken, a caprese salad, and artichoke; and we both cleaned our plates.
The presentation forms a pyramid and the preferred way of cooking is BBQ (gas or charcoal).
Grill the mushroom and chicken breast, brushing occasionally with the olive oil. You may expose them to direct, high heat at first to get some grill marks, but then cover to cook through faster. Briefly grill the sliced tomatoes as well.
Wipe lemon wedge over top of the mushroom
Stack items in a pyramid: Mushroom on the bottom, then the two slices of roma tomatoes, then crumbled feta, then the grilled chicken, then a small drizzle of the reduced balsamic, a couple twists of fresh ground pepper, a small squeeze of lemon, and finally garnish with the sprig of rosemary.