Bring a large pot of salted water to a boil and cook the pasta according to package directions until al dente. Drain and set aside, reserving about 1/4 cup of pasta water in case you need to loosen the sauce later. Cooking pasta al dente helps it hold up during baking without becoming mushy.
While pasta cooks, heat a skillet over medium heat and sauté onions and bell peppers until softened, about 3-4 minutes. Add minced garlic and cook for another 30 seconds until fragrant. Stir in shredded chicken and smoked paprika, and cook for 2-3 minutes until everything is well combined and heated through.
In a small bowl, whisk together barbecue sauce, sour cream, and milk until smooth. Season with salt and pepper to taste. This creamy BBQ sauce gives the pasta bake its signature richness and tangy flavor.
In a large mixing bowl, combine the cooked pasta, chicken and veggie mixture, and the sauce. Toss until everything is evenly coated. If the mixture feels too thick, add a splash of reserved pasta water to loosen it.
Preheat the oven to 375°F (190°C). Pour the pasta mixture into a greased baking dish and top with shredded cheddar and mozzarella cheese evenly. This cheese layer will melt beautifully and create a golden, bubbly top.
Bake uncovered for 20-25 minutes, or until the cheese is melted and slightly golden on top. Let it rest for 5 minutes before serving. Garnish with chopped parsley or green onions for a fresh, colorful touch. Serve warm and enjoy the comforting flavors with your family.