Preheat the oven to 400°F (200°C).
Heat 1 ½ tablespoons olive oil in a 10-inch ovenproof skillet over medium heat.
Add 2 tablespoons minced onion to the skillet and cook, stirring occasionally, until the onion is soft, about 2–3 minutes.
Add 1 clove garlic (minced) and ½ red bell pepper (diced) to the skillet and cook, stirring, until the pepper is softened, about 1–2 minutes.
Add 1 ½ cups roasted broccoli florets (chopped into bite-sized pieces) to the skillet and toss to combine and heat through, about 1–2 minutes.
In a medium bowl, whisk together 8 eggs, ½ cup whole milk, 1–1 ½ cups shredded cheese, 1 teaspoon kosher salt, ½ teaspoon freshly ground black pepper, and 1 teaspoon dried basil until evenly combined.
Pour the egg mixture evenly over the vegetables in the skillet and let the skillet cook on the stovetop, undisturbed, until the edges of the eggs begin to set, about 1–2 minutes.
Transfer the skillet to the preheated oven and bake until the center is set and a knife inserted near the center comes out mostly clean, about 15–17 minutes.
Remove the skillet from the oven and let the frittata rest for 5 minutes. Cut into wedges and serve warm.