Veggie Burgers: Spread 2 tsp. oil on parchment paper-lined half-sheet pan. Place mushrooms, gill side up, in pan. Drizzle with remaining oil. Sprinkle with salt and pepper.
Bake in 350°F standard oven 10 min. Turn mushrooms over and bake 10 min. longer or until tender. Remove from oven and pat off excess moisture. Cool. Pulse mushrooms in food processor to 1/4-inch pieces.
Combine mushrooms, cheese, 4.5 oz. cracker crumbs and 5.25 oz. eggs in large bowl until well blended. Form into 10 patties by pressing 3 oz. mixture into 3-1/2-inch ring mold. Refrigerate until ready to use.
Whisk cornstarch and flour in bowl. Whisk remaining eggs and water in second bowl. Place remaining cracker crumbs in third bowl. Dredge each mushroom patty in cornstarch mixture, dip in egg mixture, then coat in cracker crumbs. Fry patties in 350°F deep fryer 1 to 2 min. or until golden brown. Refrigerate until ready to use.
Honey Mustard Sauce: Mix ingredients until well blended. Refrigerate until ready to use.
For each serving: Heat 1 fried Veggie Burger in 350°F standard oven 7 to 10 min. or until internal temperature registers 165°F.
Meanwhile, brush cut sides of 1 bun with 1 tsp. butter. Toast until golden brown. Heat 2 Tbsp. onions until hot. Toss 1/4 cup arugula with 1 Tbsp. Honey Mustard Sauce.
Fill buns with arugula, Veggie Burger and onions.