Whisk together the olive oil, Dijon mustard, paprika, thyme, lemon juice and garlic in a small bowl. Sprinkle the chicken inside and out with salt and pepper. Slather the marinade all over the chicken. Place on a baking sheet, cover with plastic wrap and refrigerate for 4 hours (or up to overnight).
Remove the chicken from the fridge and let come back up to room temperature for 45 minutes.
Set your charcoal grill up for indirect heat - rack the hot the coals to one side of the grill and leave the other side bare. If you're using a gas grill, heat one side of the grill to medium-high heat and leave the other side of the grill off.
Pour off a third of the beer (or drink!), place the sprigs of thyme into the can and sit the chicken on top of the beer can. Place the chicken upright on the grill grate on the cool side of the grill, away from the heat (you may need to use a ring of heavy duty foil around the base of the can to help balance the chicken). Cover the grill with the lid and cook for 1 1/4 hours to 1 1/2 hours, rotating halfway through grilling, until the chicken is cooked through and an instant-read thermometer inserted in the thickest part of the thigh reads 165 degrees F.
Let the chicken rest for 10 minutes before carving.
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