Pour beer, paprika, chili powder, red pepper flakes, and garlic cloves into a resealable plastic bag. Add chicken, coat with the marinade, squeeze out excess air, and seal the bag. Marinate in the refrigerator, at least 4 hours.

Preheat an outdoor grill for medium-high heat and lightly oil the grate.

Cook chicken on the prepared grill until no longer pink and juices run clear, 10 to 15 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).

Meanwhile, make cheese sauce. Melt butter in a saucepan over medium heat. Whisk in flour; whisk continuously until mixture turns golden and forms a smooth paste. Pour in half and half and beer; stir until mixture thickens, 3 to 5 minutes. Stir in cheese until melted, 2 to 3 minutes more. Stir in paprika, garlic powder, and ground mustard.

Toast hamburger buns on the grill until they show grill marks, about 2 minutes. Add your favorite toppings, pour cheese sauce on top, and serve immediately.