Trim ends off green beans. Halve and juice lemon. Stem and mince parsley. Season both sides of sole fillets with ¼ tsp. salt and a pinch of pepper.
Roast the Green Beans
Combine miso and 1 Tbsp. water in a mixing bowl, add green beans and toss. Place green beans on prepared baking sheet and toss with a pinch of pepper. Spread into a single layer and roast until tender and lightly caramelized, 8-10 minutes. While green beans roast, cook fish.
Dredge the Fish
Place cornstarch on a plate. Dip sole in cornstarch, coating completely. Shake off excess and place sole on another plate.
Cook the Fish
Place a large non-stick pan over medium-high heat with 2 tsp. olive oil. Working in batches, add sole to hot pan. Cook until sole is well-browned and reaches a minimum internal temperature of 145 degrees, 1-2 minutes per side. Remove from burner.
Make the Sauce
Place a small pan over medium heat. Add butter and melt. Continue cooking until butter begins to smell nutty and turns golden brown, 2-3 minutes. Butter browns shortly after it stops sizzling, so listen closely for when it's ready. Remove from burner and add 2 tsp. lemon juice and a pinch of pepper.
Finish the Dish
Place sole on a plate and top with sauce. Serve green beans alongside. Garnish with parsley.