Preheat oven to 400 degrees F, or if you are using a gas grill, fire it up to high heat. (If your grill has a thermometer, turn between 500 and 600 degrees F.) In a medium saute pan, heat 2 Tbsp. extra virgin olive oil on medium-high heat. Once the pan is hot, add green pepper slices and saute for 5 minutes, or until soft, tossing occasionally. Add tomatoes and saute until they develop a little color and are tender. Sprinkle vegetables with salt and pepper. Toss vegetables around gently, then remove from pan and place on a paper towel to absorb oil.
Place the pizza dough onto a wide sheet of nonstick aluminum foil. Spread tomato sauce across the pizza dough, followed by 1 cup of mozzarella cheese. Arrange vegetables and basil leaves on top, followed by remaining spices and a pinch of salt and pepper. Sprinkle remaining 1/2 cup of mozzarella on top. Brush the edges of the crust lightly with olive oil.
If you are cooking in an oven... Place pizza on a cookie sheet and bake for 15 minutes.
If you are cooking on a gas grill... Once you're ready to put the pizza on, turn off one burner. Carefully place the pizza, with a wide sheet of aluminum foil underneath, on the side of the grill with the TURNED OFF burner. Close the lid on the grill, checking frequently, and let bake for about 10 minutes, or until the cheese is melted and crust is crispy.