Using a food processor, finely chop the parsley, olives, garlic and lemon zest. Pour in 1/4 cup EVOO and the lemon juice and process into a pesto.
In a large pot or deep skillet with a tight-fitting lid, heat the remaining 2 tbsp. EVOO, 2 turns of the pan, over medium-high heat. Add the onions and cook until softened, 5 to 6 minutes. Season with salt and pepper. Stir in the pesto. Add the gin and vermouth and cook until reduced, about 1 minute. Add the mussels, cover and steam until opened, 6 to 8 minutes; discard any unopened shells.
Add the butter to the mussels and shake the pan to melt it into the sauce. Serve with the bread for mopping.