Preheat your oven to 375°F (190°C).
In an oven-safe skillet, heat the olive oil over medium heat. Add the minced onion and sauté until translucent, about 3-4 minutes. Stir in the minced garlic and diced red bell pepper, cooking for another 2 minutes until fragrant.
Fold in the roasted broccoli florets, mixing well to combine with the onion and pepper. Allow the mixture to cook for about 2-3 minutes, letting the broccoli warm through.
In a mixing bowl, whisk together the eggs, whole milk, shredded cheese, kosher salt, black pepper, and dried basil until well combined.
Pour the egg mixture over the sautéed vegetables in the skillet, gently stirring to distribute the ingredients evenly. Allow the frittata to cook on the stovetop for about 5 minutes, or until the edges begin to set.
Transfer the skillet to the preheated oven and bake for 15-20 minutes, or until the frittata is puffed up and the center is set.
Once cooked, remove the frittata from the oven and let it cool for a few minutes before slicing. Serve warm or at room temperature.