Put an inverted baking sheet or pizza stone in the oven and preheat to 475 degrees F. Line another inverted baking sheet with parchment paper. Brush the dough with the olive oil and center it on the parchment. Cover with another piece of parchment; set aside.
Heat a grill pan over medium-high heat. Grill the pineapple rings until marked, about 4 minutes per side. Transfer to a plate. Grill the scallion whites, turning, until charred, about 4 minutes. Transfer to the plate with the pineapple.
Combine the barbecue sauce and tomato in a small bowl. Stretch the pizza dough to 10 by 14 inches on the parchment; spread the barbecue sauce mixture on top. Sprinkle with the pepper jack and muenster cheese and top with the grilled pineapple, ham and scallion whites and greens.
Slide the pizza (on the parchment) onto the hot baking sheet or pizza stone. Bake until the cheese is bubbling and the crust is golden, about 15 minutes. Transfer the pizza to a cutting board and slice.