Preheat the oven to 350 degrees F (180 degrees C). Brush rolls with melted butter on all sides and place on a baking sheet.
Bake in the preheated oven until lightly browned and toasted, 8 to 10 minutes. Set aside to cool slightly, about 5 minutes.
While rolls bake, whisk pudding mix, milk, vanilla, and salt together in a bowl until well combined and thickened, about 2 minutes. Refrigerate, uncovered, for 10 minutes to set up.
Transfer filling to a piping bag or a plastic bag. Make a small cut in the corner if using a plastic bag.
Use a paring knife to make a small cut into one side of each roll. Push knife up and around to create an about 3/4-inch square opening without going all the way through. Fill each hole with about 1 1/2 tablespoons filling.
Spread frosting evenly over tops of filled donuts, about 4 teaspoons each. Place sprinkles in a medium bowl. Dip top of each filled roll in sprinkles, ensuring frosting is fully covered by sprinkles. Serve immediately. Recipe developed by Melissa Gray-Streett
