Preheat the oven to 400 degrees F. Place the butter in a medium-sized skillet and set on the stovetop over medium heat. Add the onions and sauté for 2 minutes. Then stir in the chopped red pepper and sauté another 2 minutes. Turn off the heat.
Pour the tortellini in a large 3-quart (9x13-inch) baking dish. Place the chopped chicken, fresh spinach, and sautéed onions and peppers in the dish. Toss to mix evenly and spread out in the pan.
Mix the pesto, chicken broth, heavy cream, 1 teaspoon salt, and 1/2 teaspoon ground black pepper in a bowl or measuring pitcher. Whisk well, then pour the mixture over the tortellini.
Cover the pan with foil and bake for 20 minutes. Then gently stir the pasta to make sure the tortellini on top is covered in sauce. Sprinkle the top with both cheeses and bake, uncovered, for another 15-20 minutes. Serve warm.
NOTE: 1. This recipe can easily be halved and baked in an 8- or 9-inch square pan for a smaller batch. 2. You can add the onions and peppers to the baking dish raw for a true one-pan recipe, but I feel sautéing them first improves the flavor and texture.