Wash and dry all produce. Preheat oven to 450 degrees. Cut potatoes into ½-inch cubes. Place on baking sheet and toss with a large drizzle of olive oil. Roast until crispy on outside and tender on inside, 25-30 minutes.
Trim green beans. After potatoes have roasted for 10 minutes, toss green beans with a drizzle of olive oil in a medium bowl. Add to baking sheet with potatoes and roast until both are tender, 15-20 minutes.
Meanwhile, mince or grate 1 clove garlic (reserve remaining clove for another use). Finely chop parsley. Cut lemon into wedges. Drain capers. Pat sole dry with a paper towel. Season all over with salt and pepper. Dust fillets evenly with flour on both sides.
Heat a drizzle of olive oil in a large pan over high heat. (TIP: Grab a nonstick pan if you have one.) Add sole to pan and cook until firm, opaque, and lightly browned, 2-3 minutes per side. Halfway through, add ½ TBSP butter to pan to help brown. Remove sole from pan and set aside.
Reduce heat to medium and add garlic to pan. Cook until lightly browned, about 1 minute. Stir in sour cream, capers, half the parsley, 2 TBSP water, and a squeeze of lemon juice. Continue stirring until slightly thickened, about 1 minute. Remove pan from heat.
Serve fish on a bed of green beans and potatoes. Drizzle with sauce. Garnish with remaining parsley. Serve with lemon wedges on the side.