Preheat the oven to 400°F (200°C).
Melt 1 tablespoon butter in a large skillet over medium heat, then add the chopped red onion and sauté 2 minutes.
Add the chopped red bell pepper to the skillet and sauté another 2 minutes, then remove from heat.
Place the tortellini in a 9x13-inch (3-quart) baking dish. Add the cubed chicken, fresh spinach, and the sautéed onions and peppers; toss to combine and spread evenly.
In a bowl or measuring pitcher, whisk together 6 ounces basil pesto, 2 cups chicken broth, 1 cup heavy cream, 1 teaspoon salt, and 1/2 teaspoon ground black pepper until smooth.
Pour the pesto-cream mixture evenly over the tortellini and chicken in the baking dish.
Cover the dish with foil and bake for 20 minutes, then carefully stir the casserole to redistribute sauce.
Sprinkle the top with 2 cups shredded fontina and 1/2 cup shaved Parmesan, uncover, and bake another 15–20 minutes until the cheese is melted and bubbly.
Let rest a few minutes, then serve warm.