Preheat the oven to 400 degrees . Butter the toasted bread while hot, then finely chop.
In a medium skillet, melt 3 tbsp. butter over medium heat. Add the celery, chile, shallot, garlic and thyme; season with salt and pepper. Cook until softened, then stir in the vermouth to deglaze. Transfer the mixture to a bowl to cool. Add the crabmeat, chopped bread, lemon juice, Old Bay and hot sauce to the bowl; mix well with your hands.
Place the shrimp butterflied side up on a greased baking sheet. Mound the crab mixture at the base of the butterflied shrimp and rest the tail over the top of the stuffing. Bake until the shrimp are pink and firm, 12 to 15 minutes.
Meanwhile, make the beurre blanc: In a small saucepan, bring the vinegar and vermouth to a simmer over low heat. Add the shallot and season with salt and pepper. Simmer until reduced by half, 2 to 3 minutes. Remove the pan from the heat and immediately whisk in 4 tbsp. butter until incorporated. Return the pan to low heat and whisk in the remaining 12 tbsp. butter, 1 piece at a time. Sir in the lemon juice, then the chives.
To serve, place a puddle of beurre blanc on plates or in shallow bowls. Arrange 4 shrimp in the sauce, with the tails toward the center.