Preheat oven to 500 degrees. Spray a large baking sheet with nonstick cooking spray.
Place tomatoes in a colander and drain as much liquid as possible.
Combine tomatoes, olives, capers, salt, and crushed red pepper flakes in a large bow. Mix well.
Stir mozzarella and feta cheeses into the tomato mixture. Then add broccoli to bowl and combine well. Set aside.
Sprinkle your work surface with a little flour, then divide dough into 4 equal pieces.
Lightly sprinkle a piece of dough with flour, then gently stretch it into an 8-inch circle. (Tip: if your dough doesn't want to stretch, let the dough rest on your work surface for a few minutes, then continue stretching.)
Top one half of the dough circle with ¼ of the broccoli mixture.
Use your fingers to apply a little water around the edges of the filling side of the dough.
Carefully fold the dough over the filling, then press the edges together. Go around the edge with a fork to seal the dough together.
Transfer the calzone to the baking sheet, then use a sharp knife to cut 3 small slits in the top of the calzone.
Brush calzone lightly with olive oil. Repeat with remaining dough and filling to make 4 calzones.
Bake calzones for 15 minutes or until browned all over.