Wash and dry all produce. Adjust broiler rack so that it is in position closest to flame and place a baking sheet on rack. Preheat broiler to high. Core and dice tomato. Thinly slice scallions, separating greens and whites. Tear pulled pork into larger bite-size pieces.

Heat 1 TBSP butter and 1 TBSP olive oil in a large pan over medium-high heat. Stir in barbecue seasoning. Cook until fragrant, about 30 seconds.

Add pulled pork and 1/4 cup water to pan, stirring to coat. Cook, stirring occasionally, until warmed through and water is mostly evaporated but still a little saucy, 2-3 minutes. Season with salt and pepper.

Scatter pork evenly over flatbreads, then repeat with tomato, scallion whites, and mozzarella. Carefully remove baking sheet from broiler and sprinkle with a drizzle of olive oil. (TIP: Get out your mitts; the sheet will be hot.) Transfer flatbreads to sheet and sprinkle each with a large drizzle of olive oil.

Return baking sheet to broiler and broil until cheese melts and flatbreads begin to turn golden brown and crisp, 3-4 minutes. TIP: Keep an eye on these; they can burn easily.

Drizzle barbecue sauce over flatbreads and sprinkle with scallion greens. Cut into slices and serve.