Melt butter in a large skillet over medium heat. Stir in flour; cook until golden, about 2 minutes. Add chicken broth, almond milk, and cream; cook and stir for 2 minutes.
Stir in red pepper flakes, garlic, onion powder, salt, and pepper. Increase heat and bring to a boil, stirring constantly. Reduce heat to low and cook for 3 minutes. Add Mexican, Italian, and Cheddar cheeses. Cook and stir until melted, 2 to 5 minutes.
Turn heat to high, add broccoli, and bring to a boil. Reduce heat to medium and cook for 5 minutes. Increase heat to high again, and cook until broccoli is tender but not mushy, 2 to 3 minutes.