Prepare the sauce: In a skillet, add olive oil, diced onion, diced bell peppers and shredded carrots, cooking over medium heat, stirring often until vegetables are softened.
Stir in tomato sauce, brown sugar, basil, thyme, oregano, salt and pepper.
Cook for 5 minutes, remove from heat while you prepare the dough.
Prepare a grill fire to about 450° with a pizza stone and hickory for smoke flavor.
Make the pizza dough, rolling pizza into about 3, 10- 12 inch circles.
Fold the edges over to make the sides.
Sprinkle a pizza peel or pan with cornmeal.
Carefully place the dough over the cornmeal.
Spread pizza sauce over dough in a thin, even layer.
Sprinkle chopped brisket over the sauce.
Top with mozzarella cheese.
Next, add the thin sliced tomatoes and green onions.
Sprinkle the pizza stone with cornmeal.
Slide the pizza onto the stone and close the lid.
Lower grill to about 350° and cook for about 3- 5 minutes.
Use a large spatula to turn pizza a 1/4 turn and continue to cook with the grill closed.
Continue to rotate pizza about every 3- 5 minutes, cooking until pizza is well browned on the bottom and the cheese starts to bubble. Total cooking time should be about 15- 20 minutes.
Remove from grill and cool 5 minutes before slicing.