Preheat grill for medium heat and lightly oil the grate.
Mix chicken and 2 tablespoons barbeque sauce together in a bowl until evenly coated.
Roll dough onto a work surface and cut in half to form 2 rectangles. Brush the top of each rectangle with olive oil.
Place dough, oil-side down, onto the grill; cook until bottom of dough is browned and top is bubbling up, about 5 minutes. Brush the tops of each dough with oil and flip.
Spread the remaining barbeque sauce over each crust; top with chicken, Colby-Monterey Jack cheese, and tomatoes.
Close the lid of the grill and cook pizza until cheese is melted, 2 to 3 minutes. Remove pizza from grill and cool before slicing, 2 to 3 minutes.
