Position a rack in the upper third of the oven and heat the oven to 400°F.
Melt 4 Tbs. of the butter in a medium saucepan or skillet over medium-low heat. Lightly brush the inside of a 9x13-inch Pyrex baking dish with some of the melted butter. Transfer 1 Tbs. of the melted butter to a small dish and keep warm. Add the scallions to the remaining butter in the saucepan and cook, stirring, until softened, about 2 min.
In a medium bowl, mix the melted scallion butter with the breadcrumbs, tarragon, beaten egg, 1/4 tsp. salt, and several grinds of pepper.
Lightly season the fillets on both sides with salt and pepper. Lay a fillet on a work surface with the pointed end facing you. Mound a quarter of the stuffing on the pointed half of the fillet. Fold the wide, split end of the fillet over and press firmly (but don’t squish) to cover the stuffing. Some stuffing will remain exposed. Put the stuffed fillet in the buttered baking dish. Repeat with the remaining fillets. Brush the tops of the fillets with the reserved tablespoon of melted butter. Pour the vermouth and chicken broth around, not over, the fillets.
Bake until the fish feels firm but flaky and the insides of the fillets look opaque when pierced with the tip of a sharp knife, 20 to 25 min. Using a thin, flexible slotted metal spatula, transfer the fillets to four dinner plates.
Add 1 Tbs. cold butter to the hot baking dish and whisk until the butter melts. Repeat with the remaining 1 Tbs. butter and the mustard. Season the sauce with salt and pepper and spoon over the fillets. Serve immediately.