Preheat the broiler. Bring a pot of salted water to a boil. Add the pasta and cook as the label directs; drain and set aside.
Meanwhile, season the chicken with salt and pepper. Heat 2 teaspoons olive oil in a large nonstick skillet over medium-highheat. Add the chicken and cook until golden and cooked through, 4 minutes per side. Transfer to a baking sheet.
Add the red peppers and oregano to the skillet; cook, flipping the peppers, until warmed through, about 2 minutes. Lay a pepper half on each piece of chicken. Top each with a slice of mozzarella.
Add the remaining 1 tablespoon plus 1 teaspoon olive oil and the garlic to the skillet; cook until softened, 1 minute. Add the kale and cook until slightly wilted, 2 minutes. Add the broth and bring to a boil; cook until the kale is tender, 4 to 5 minutes. Add the pasta and parmesan; season with salt and pepper. Cook, tossing, until warmed through.
Transfer the chicken to the broiler and broil until the cheese melts, about 1 minute. Serve with the kale pasta.