Place a foil-lined baking sheet (2 baking sheets for 4 servings) on top rack and preheat oven to 450 degrees. Wash and dry all produce. Halve, peel, and thinly slice onion. Drain pineapple over a small bowl, reserving juice. Stir 1/4 of the sliced onion, half the vinegar (you’ll use the rest later), and a pinch of salt into bowl with pineapple 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 necessary to prevent sticking, until softened, 6-8 minutes.

Meanwhile, halve, core, and deseed poblano; slice crosswise into very thin strips. Toss in a second small bowl with a drizzle of olive oil, salt, and pepper. Dice mozzarella into 1/2-inch pieces. Finely chop cilantro leaves and tender stems.

Once onion is softened, increase heat under pan to medium high. Add another drizzle of oil and pineapple chunks; cook, stirring, until pineapple is lightly browned, 2-3 minutes. Stir in remaining vinegar. Cook, stirring, until jammy and caramelized, 1-2 minutes more. Turn off heat and season with salt and pepper.

Brush or rub each side of flatbreads with a drizzle of oil. Remove heated baking sheet from oven; place flatbreads on it (for 4 servings, divide flatbreads between 2 baking sheets). Evenly spread with BBQ sauce, then top with caramelized onion and pineapple, poblano, and mozzarella.

Return flatbreads to top rack and bake until cheese is melted and edges are golden brown, 12-15 minutes. (For 4 servings, rotate baking sheets halfway through baking.) Transfer to a cutting board and top with pickled onion and cilantro. Cut into slices and serve.