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adjust broiler rack so that it is in position closest to flame and place a baking sheet on rack. preheat broiler to high.
heat butter and 1 tbsp olive oil in a large pan over medium-high heat. stir in bbq seasoning. cook until fragrant, about 30 seconds.
add pulled pork and ¼ 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. 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 crips, 3-4 minutes.
drizzle bbq sauce over flatbreads and sprinkle with scallion greens. cut into slices and serve.