Step 1Stand peppers on cutting board; cut off sides in 4 slabs, following natural contours as much as possible. Arrange on baking sheet lined with foil, skin side up. Broil on rack 6 inches from heat source until skin blackens, 4 to 6 minutes. Remove and wrap in foil lining. Set aside to steam 10 minutes, then slip off skins.
Step 2Spray both sides of mushrooms and eggplant with olive oil and generously sprinkle with salt and pepper. Put on 2 baking sheets lined with foil.
Step 3Bake mushrooms at 400 degrees 4 minutes, turn and bake 4 minutes more. Set aside. Bake eggplant slices at 400 degrees 6 minutes, turn and bake 6 minutes more. Trim eggplant slices to match size of mushrooms. Set aside. Reline 1 baking sheet with foil.
Step 4Assemble burgers in this order: 1 mushroom (cut-side up), 2 tablespoons goat cheese, onion, salt, pepper, eggplant, red pepper, onion, 2 tablespoons goat cheese, 2 basil leaves. Top with mushroom cap (cut-side down). Set on baking sheet and cover with foil. (Can be made several hours ahead and kept at room temperature.) Bake until warm, about 5 minutes.