Rinse the mussels under cold running water while scrubbing with a vegetable brush. Remove the stringy mussel beards with your thumb and index finger as you wash them. Discard any mussels with broken shells.
Heat 1 tablespoon olive oil and 1 tablespoon butter in a large pot over medium heat. Add the leeks, garlic, and thyme and cook until the vegetables cook down to a pulp, about 5 minutes. Add the mussels and give everything a good toss. Add the white wine. Cover and steam over medium-high heat for 10 minutes, until the mussels open. Stir occasionally so that all the mussels are in contact with the heat. Add the remaining 2 tablespoons butter and a drizzle of olive oil to the sauce remaining in the pot and swirl to incorporate. Season with salt and pepper. Sprinkle with the parsley and serve immediately.
Recipe courtesy of Tyler Florence