Adjust rack to top position (top and middle positions for 4 servings) and preheat oven to 450 degrees. Line a baking sheet (two baking sheets for 4) with foil and place on top rack (top and middle racks for 4). Wash and dry produce.

Halve, peel, and thinly slice onion. Drain pineapple over a small bowl, reserving juice. Stir one-quarter of the onion, half the vinegar (you’ll use the rest later), and a pinch of salt into bowl with juice. Set aside.

Heat a drizzle of oil in a large pan over medium heat. Add remaining onion and cook, stirring and adding splashes of water as needed to prevent sticking, until softened, 6-8 minutes.

While onion cooks, halve, core, and very thinly slice green pepper into strips. Toss in a second small bowl with a small drizzle of olive oil, salt, and pepper. Dice mozzarella into ½-inch pieces.

Once cooked onion is softened, increase heat under pan to medium high. Add pineapple; cook, stirring, until pineapple is lightly browned, 2-3 minutes.

Stir in remaining vinegar. Cook, stirring, until onion and pineapple are caramelized and jammy, 1-2 minutes more. Turn off heat and season with salt and pepper.

Brush or rub each flatbread with a small drizzle of oil. Remove prepared baking sheet from oven; carefully place flatbreads on sheet. (For 4 servings, divide flatbreads between both prepared sheets.)

Evenly spread flatbreads with BBQ sauce, then top with caramelized onion and pineapple, green pepper, Monterey Jack, and mozzarella.

Bake flatbreads on top rack until cheese melts and edges are golden brown, 10-12 minutes. (For 4 servings, bake on top and middle racks; swap rack positions halfway through baking.)

Transfer flatbreads to a cutting board and top with pickled onion (draining first) to taste. Slice into quarters, divide between plates, and serve.