For the Green Olive Sauce, put everything into a food processor and process until almost smooth. Taste and adjust seasoning with salt and pepper. Set aside.
Rinse the mussels and check them over; discard any that do not close when tapped. In a large heavy bottomed pot with a tight fitting lid, heat the olive oil over medium-high heat. Add the shallots, garlic, chile, lemon, and thyme and cook for about 5 minutes. Add the mussels and give them a stir. Pour in the white wine, cover, and cook, shaking the pot occasionally, until all the mussels have opened, about 10 minutes. Stir in the Green Olive Sauce, pour the mussels onto a platter, and serve.