Put 1/4 cup olive oil, the lemon juice and zest, garlic, sugar, and salt and pepper to taste in a blender and pulse to combine.
Preheat a gas or charcoal grill to medium (you can hold your hand 1 to 2 in. above cooking grate only 4 to 5 seconds). Brush oil on both sides of bread and put on grill; sprinkle with salt and pepper. Grill 2 minutes. Turn over, top with cheese, close lid, and grill until cheese melts, 1 to 2 minutes.
Top cheese evenly with ham. In a large bowl, toss arugula with vinaigrette and arrange a handful on each sandwich. Drizzle remaining vinaigrette over sandwiches.
Note: Nutritional analysis is per sandwich.