Cut sweet potato into ½" thick fries. Peel and slice red onion into ¼” thick rounds. Cut mushrooms into ¼” slices. Rinsing starch off cut potatoes helps fries crisp up in oven.

Place sweet potato fries on prepared baking sheet and toss with 2 Tbsp. olive oil and 1 tsp. salt. Spread into a single layer and bake until browned and tender, 25-30 minutes, turning after 15 minutes. Rest 5 minutes on baking sheet to firm up after cooking. Combine mayonnaise, 2 tsp. miso (reserve remaining for burgers), and sun-dried tomato pesto in a small bowl and set aside.

Combine ground beef, remaining miso, half the Parmesan (reserve remaining for fries), 1 tsp. salt, and 1 tsp. pepper in a medium mixing bowl. Form four burgers 5” in diameter. Using your fingers, make a small dimple in middle of each burger. This will prevent patties from bulging when cooked and create an evenly flat burger.

Heat a large non-stick pan over medium-high heat. Brush cut side of buns with 1 Tbsp. olive oil and toast, cut side down, until golden brown, 1-2 minutes. Reserve pan for cooking burgers (no need to wipe clean).

Return pan to medium-high heat and add 2 tsp. olive oil and burgers to hot pan. Cook on first side until brown, 3-4 minutes. Flip burgers and add mushrooms. Cook until mushrooms are brown and burgers reach a minimum internal temperature of 145 degrees, 4-5 minutes. Transfer burgers to a plate and top with sliced white cheddar. Depending on how well you like your burgers done, you may end up removing them before mushrooms have finished browning.

Arrange sweet potato fries on plate and garnish with remaining Parmesan. Place a bottom bun on plate and top with tomato-miso aioli, burger, mushrooms, and red onions. Cover with top bun.