In a food processor, whirl garlic to finely chop. Add butter, mustard, parsley, lemon juice, kosher salt, and pepper; whirl until well blended. Transfer to a bowl.
Meanwhile, boil a quarter of beans at a time in a large pot of water until barely tender, about 3 minutes. Using tongs, immediately transfer beans to ice water and let cool, then transfer to a colander to drain.
Preheat broiler with 1 rack 5 to 6 in. from heat and another rack below it. Blot beans dry, then arrange lengthwise in 2 rimmed baking pans. Dot beans in each pan with about a quarter of butter.
Cook beans in oven 1 minute. Toss to coat with butter and switch positions of pans. Cook until top beans are blistered and speckled brown, 4 to 9 minutes. Remove top pan from oven, then broil second pan the same way.
Transfer beans to a platter and sprinkle with coarse sea salt. Serve with remaining butter if you like.
Make ahead: Prepare recipe through step 2; chill butter and beans airtight up to 1 day.
Note: Nutritional analysis is per serving.