Preheat barbecue or broiler.

Place olive oil and juice and zest of 3 lemons into a small saucepan and place over medium heat. Bring to a near boil, remove from heat and pour into a bowl. Immediately add the marjoram "brush" and limoncello and allow to steep, like tea, for 1 hour, covered. This mixture can be stored in a lidded jar, away from light, for up to a week.

Kill lobsters by piercing their shells with a sharp point at the center of the back end of the eyes. Place the whole lobster on the hottest part of the grill for 3 minutes per side and remove from the heat. Cut the lobsters in half lengthwise and, careful not to lose a drop of the tomalley or roe, gently anoint them with the scented oil, using the marjoram brush. Gently place them, shell-side down, on the grill and cook until nearly done, about 6 to 7 minutes. Turn the flesh side down, cook for 1 minute, and remove to a platter. Dress the arugula with 4 tablespoons of the scented oil and some coarse salt and pile in center of platter. Serve warm or room temperature, with the remaining oil and lemon wedges on side.