Preheat oven to 350 degrees F.
Heat 2 tablespoons olive oil in a large nonstick saute pan over medium heat. Once the oil is hot, add the broccoli, onions, red pepper flakes, and 2 cloves minced garlic. Slowly saute the broccoli, stirring occasionally until it starts to brown all over, about 10 minutes. Add the olives and continue to cook about 1 more minute. Remove from heat and add the ricotta cheese, stirring to evenly distribute.
While broccoli and onions are caramelizing, prepare the tomato sauce: heat 1 teaspoon olive oil in a medium-size saucepan over medium heat. Add remaining 2 cloves minced garlic and cook for 1 minute. Add the tomato paste and stewed tomatoes. Bring to a simmer and cook for 5 minutes. Stir in the basil and remove from the heat.
Spread about 1/3 of the sauce in the bottom of a 13 by 9 by 2 inch-baking dish. Stuff the shells with the broccoli mixture. Arrange the stuffed shells in the baking dish and top with the remaining sauce. Sprinkle evenly with the mozzarella. Bake, uncovered, for 20 minutes. Serve immediately.