Cook garlic in butter in a large frying pan over medium-high heat until fragrant. Stir in mushrooms and yellow onion and sauté until mushrooms are nicely browned, about 8 minutes. Season with 1/4 tsp. salt and 1/8 tsp. pepper. Set aside.
Mix together remaining salt and pepper, turkey, red onion, herbs, and 1/4 cup mayonnaise in a bowl. Shape into 16 patties about 3 in. wide, pressing a dimple into middle of each (this helps them cook evenly).
Heat grill to high (450° to 550°). Brown rolls on cut sides until toasted, about 1 minute. Oil cooking grates, using a wad of oiled paper towels and tongs. Cook patties until grill marks appear on both sides, turning once, 5 to 6 minutes total.
Reheat mushroom mixture over high heat and assemble burgers: about 1 tbsp. cranberry mustard on bottom of each roll, then arugula, patties, and mushroom mixture. Spread tops of rolls with a little mayonnaise and set on burgers.
Note: Nutritional analysis is per 2-slider serving.