Preheat and set up the grill for indirect heat, stabilizing the temperature between 350°F and 375°F (175°C–190°C).

In a small bowl, mix paprika, brown sugar, garlic powder, onion powder, cayenne pepper, black pepper, salt, dried thyme, and dried oregano to create the spice rub.

Open the beer can and pour out (or drink) about 1/4 cup to create headspace for steaming.

Pat the chicken completely dry with paper towels. Rub the skin with vegetable oil, then coat the bird inside and out with the prepared spice rub, working some rub under the skin where possible.

Stuff the chicken cavity with the lemon halves and smashed garlic cloves.

Slide the chicken cavity over the beer can and stand the bird upright on a beer can holder or a sturdy foil tripod.

Place the upright chicken on the cool side of the grill (indirect heat), close the lid, and cook for about 1 hour 15 minutes, maintaining 350°F–375°F (175°C–190°C).

Check doneness by inserting an instant-read thermometer into the thickest part of the thigh without touching bone; the chicken is ready when it reaches 165°F (74°C).

Carefully remove the chicken from the grill, tent loosely with foil, and let it rest for 10 minutes before carving and serving.