Combine the paprika, kosher salt, black pepper, onion powder, garlic powder, dried thyme, and cayenne pepper in a small bowl. Mix until well blended.
Pat the chicken completely dry with paper towels. Rub olive oil all over the bird, including under the skin where possible.
Season the chicken generously with the spice rub, making sure to coat all surfaces evenly.
Open the beer can and pour out about half of it. Poke a few extra holes in the top of the can.
Carefully lower the chicken cavity over the beer can so it sits upright. The two legs and the beer can should form a stable tripod. Tuck the wing tips behind the back.
Preheat the grill to medium heat (350-375°F). Place the chicken in the center over indirect heat. Close the lid and grill for about 1 hour and 15 minutes, or until a thermometer inserted into the thickest part of the thigh reads 165°F.
Carefully transfer the chicken and can to a cutting board. Let it rest for 10 minutes loosely covered with foil, then carefully remove the beer can and carve.