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.