Make filling: Heat 1 tablespoon butter and oil in a large skillet over medium heat. Cook and stir chicken in hot butter-oil mixture until no longer pink in the center, about 5 minutes. Transfer to a plate.

Melt remaining 4 tablespoons butter in the same skillet over medium heat. Add mushrooms and onion; cook and stir until softened, about 5 minutes. Stir in flour and cook for 2 minutes. Reduce heat to low. Add milk, chicken bouillon cubes, salt, and pepper; cook and stir until thickened into a sauce, about 5 minutes. Stir in sour cream and sherry until sauce is smooth.

Scoop 1 cup sauce into a small bowl; keep warm. Stir chicken and 1 teaspoon parsley into remaining sauce in the skillet. Simmer until chicken is heated through, about 5 minutes.

Make crêpes: Beat together milk, flour, egg, melted butter, sugar, vanilla, and baking powder in a large bowl until well combined.

Grease a 7-inch skillet lightly; heat over medium heat. Pour 2 tablespoons batter into the hot skillet, swirling to spread evenly. Cook crêpe until golden brown, about 30 seconds per side. Transfer crêpe to a plate. Repeat with remaining batter, separating crêpes with waxed paper and covering to keep warm.

Scoop 1/4 cup chicken mixture onto each crêpe and roll up. Spoon reserved 1 cup sauce over filled crêpes. Garnish with remaining 1 teaspoon parsley.