Whisk together the lemon juice, olive oil, mustard, rosemary, garlic and a big pinch of salt and pepper in a measuring cup. Add chicken to a gallon-size zip top bag or casserole dish and pour the marinade over the chicken. Turn to coat and let the chicken marinate at room temperature for 30 minutes (or 1 hour in the refrigerator).

Heat a grill to medium-high. Brush the grill grates with olive oil using a clean tea towel or paper towel.

Remove the chicken from the marinade and discard the marinade. Place the chicken on the grill and cook until golden and slightly charred, about 3 minutes per side. Place the lemon halves cut-side down on the grill grate and grill until charred, about 4 minutes.

Add the arugula to a platter and drizzle very lightly with olive oil. Remove the chicken from the grill and shingle across the bed of arugula. Squeeze some lemon juice over the chicken and serve.