Place flour in a shallow dish or resealable plastic bag and season with salt and pepper. Dredge fillets in flour mixture or add to bag, close, and shake to coat.
Place 1 tablespoon of the butter and oil in a large frying pan over medium heat. When it foams, lower heat to medium-low. Shake excess flour off 2 of the fillets, add to pan, and cook until golden brown and crisp, about 2 minutes per side. Remove fillets to a plate and repeat with remaining fillets.
Add remaining 1 tablespoon butter and cook until it begins to brown. Immediately remove from heat, stir in lemon juice, and pour over fish. Lay a lemon slice over each fillet, sprinkle pistachios over top and serve with mashed or roasted potatoes and a simple green salad.