Hollow out a shallow depression in each mushroom on the gill side. In a large resealable bag, combine the garlic, lemon zest and juice, shallots, 1/2 teaspoon salt and 1/4 teaspoon pepper with the mushrooms. Refrigerate for 45 minutes to 1 hour.
Heat a grill to medium-high heat; brush the grill grates with vegetable oil. Place the marinated mushrooms on the grates stem/gill-side down and grill for 6 to 7 minutes. Flip and crack 1 egg on top of each mushroom in the hollowed-out depression. Close the lid and grill until the egg whites are solid but the yolks are still runny, 6 to 7 minutes.
Whisk together the olive oil, rice wine vinegar, honey mustard and some salt and pepper in a bowl. Add the arugula and tomatoes and toss together.
Top the mushrooms with the arugula salad, shaved aged Cheddar and cracked black pepper and serve.
Recipe courtesy of Jeff Mauro