In a small bowl, stir together the butter, basil, garlic, onion, dijon, parmesan, flour, and lemon juice. It'll be like a weird butter paste.
In a large skillet, heat the oil over medium-high heat. Season your fish with a little salt and pepper and lay it in the hot oil. Cook for about 4 minutes.
Flip your fish to cook the other side. Top each fillet with a large spoonful of the butter mixture. Cook another 4 minutes. Actual cooking time will depend on the thickness of your fish. The fish is done when it flakes easily with a fork.