Bring a large pot of salted water to a boil. Cook the spaghetti until al dente, according to package directions. Drain and set aside.
Season both sides of the chicken with smoked paprika, garlic powder, salt, and pepper.
To a large skillet over medium-high heat, add 1 tablespoon of olive oil. Once hot and shimmering, add the chicken and cook for about 3 to 4 minutes per side, until golden and fully cooked through to an internal temperature of 165°F. (Cooking times may vary, depending on the size of the chicken breasts.) Transfer to a plate and cover to keep warm.
Reduce heat to medium. Add the remaining 1 tablespoon of olive oil. Stir in the minced garlic and crushed red pepper flakes, and cook for 30 seconds.
Slowly pour in the chicken broth and use a spatula to scrape up any browned bits from the bottom of the pan.
Stir in the sour cream and cream of chicken soup. Whisk until the mixture is completely smooth and combined. Bring to a gentle simmer, stirring occasionally.
Stir in 2 cups of the Monterey Jack cheese and all of the sharp cheddar cheese. Continue stirring until the cheese is melted and the sauce is smooth.
Add the spinach and stir until wilted, about 1 to 2 minutes.
Add the cooked spaghetti and toss well to coat, making sure the noodles are evenly distributed and fully coated in the sauce.
Place the cooked chicken breasts on top of the spaghetti. Sprinkle with the remaining Monterey Jack cheese.
Top with crispy onions. Broil on low for 2 minutes, or until the cheese is melted and bubbly and the onions are crispy.
Let rest for 5 minutes before serving.