Roll the puff pastry into a rectangle 11 x 15 inches. Transfer it to a 11 x 17-inch sheet tray lined with parchment paper and fit the dough to the tray exactly. Be sure the dough is evenly rolled - about 3/8-inch thick. Freeze the sheet tray for about 1/2 hour.
Remove the tray from the freezer and using a fluted 2 1/4-inch round cutter, cut out 24 to 28 circles. Using a fork, dock each round well in the center, 4 to 6 times; leave a 1/4-inch rim undocked. The circles can be put back in the freezer for up to 2 months. Once they're frozen, they can be stacked and wrapped in plastic wrap.
Preheat oven to 425 degrees F. Heat some oil in a saute pan and cook the onions slowly, until they are soft and fully cooked. Season with salt and pepper. Set aside to cool. Slice the tomatoes about 3/8-inch thick. Cut off broccoli flowers about 1 1/4 inches around. Cut them as close to the flower as possible, without them falling apart. Cut the stems off the mushrooms flush with the bottom of the cap (don't hollow them) and slice as thinly as possible. Fill a little bowl with about 1-inch of oil.
Place the rounds of dough, upside down on a papered cookie sheet and dock again. Brush about 1/3 with oil, spread about 1/3 with a little olive paste or pesto, and spoon about 1/2 of the cooled onions on the remaining rounds. Sprinkle each with Parmesan.
Place a tomato slice on the rounds spread with oil only; the slice should be the same size or a little smaller than the round of dough. Garnish with a few strips of fresh basil.
Dip each broccoli flower into the oil, head down, and press it, flower up, into the center of each round spread with olive paste.
Press a sliced mushroom cap into the center of each onion round. Slant the cap a little so that the slices separate.
Brush each round generously with oil and sprinkle generously with Parmesan, salt and pepper. Bake for about 17 minutes or until the bottoms are lightly browned. Serve warm.