Combine the lemon juice, zest, mustard, rosemary and crushed red pepper in a small bowl. Add 1 to 2 tablespoons of olive oil to the mustard mixture to loosen it. Paint this mixture generously all over the chicken and let sit for up to 2 hours at room temperature or overnight in the refrigerator. If you have any of the mustard mixture left over, save it for repainting while the chicken is cooking.
Preheat the grill to medium (you want the chicken to cook on the grill as long as possible without burning to develop lovely crispy, tasty skin). Brush off any crud on the grill with a wire grill brush and run an oiled towel over the grill to pick up any bits of loosened crud. Season the chicken lightly with salt and arrange gently on the grill, skin side down, and cook for 5 to 6 minutes. If the chicken flares up, move it to a cooler part of the grill; you don't want sooty film on the chicken (it won't taste good).
Once the chicken has started to brown and crisp, it's a good idea to close the grill to allow the heat to cook the chicken from all directions. You want to cook the chicken skin side down for a total of 10 to 12 minutes, then turn it over and cook it for another 10 to 12 minutes. If you're using a grill pan, you can transfer the chicken to a 375 degree F oven when it's ready to flip so it can finish cooking. If there is leftover mustard mixture, use it to paint the skin of the chicken on the second side. The chicken is done when an instant-read thermometer registers 165 degrees F.
When the chicken is nearly done, place the lemon halves on the grill cut side down and leave them alone until brown and caramelized, 5 to 6 minutes. (These will smell great while they're grilling!)
Remove the chicken from the heat and serve with the grilled lemon for extra lemony goodness.
During the cooking time it is a great idea to close the grill to allow the heat to cook from all directions. If grilling indoors, the chicken can be transferred to a preheated 375 degree F oven to finish cooking once the grill marks have been established.
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