Pat the fish dry with paper towels, and season both sides of the fish with salt and pepper.
Place the almond flour on a large plate. Dredge the sole on both sides in the almond flour.
In a large nonstick skillet over medium-high heat, heat the olive oil and half of the butter. Swirl the skillet until the butter starts to foam and brown flecks appear, about 1 minute. Reduce the heat to medium to keep the fat sizzling but not burning, then add the fish fillets and fry until browned on each side, 2-1/2 minutes per side, 5 minutes total. Transfer the fish to 2 dinner plates, reserving the skillet.
Meanwhile, in a medium pot with a lid, bring to a boil 1 inch of water and 1/2 teaspoon salt. Add the green beans, cover, and let steam for 4 minutes. Drain, then divide the beans between dinner plates.
Add the remaining butter and the lemon juice to the reserved skillet, whisk over medium heat for 2 minutes, then stir in the parsley. Pour the sauce over the fish and green beans. Enjoy!