Combine 1 cup of flour, sugar, instant yeast, and salt. Add butter, egg, and milk, and whisk the batter vigorously until smooth.
Add the remaining flour and continue mixing until well combined. The dough will be quite sticky. Transfer to a well-floured surface and knead for about five minutes or until it's no longer sticky and appears relatively smooth. You may need to add up to ¼ cup of flour as needed. Shape the dough into a ball and place it back in the bowl. Loosely cover the bowl with plastic wrap and let it rise in a warm spot for 1 hour.
Once the dough has doubled in size, transfer it to a well-floured surface and flatten it to a ¾-inch thick rectangle. Use a dough scraper to cut the rectangle into logs, then divide each log into small pieces.
Roll each piece into a ball and place it on a parchment-lined baking sheet. Leave about 1 inch of space between each ball. Loosely cover with plastic wrap and let it rise in a warm spot for 1 hour.
Heat 1 inch of oil to 285℉ in a high-walled pot. Fry 6-8 Timbits at a time for 2-3 minutes, gently stirring and turning them constantly, until golden brown on all sides. Transfer to a paper towel-lined plate to dry.
In a medium bowl, whisk milk, confectioner's sugar, and vanilla extract until it has the texture of thin paint. If it's too thick, add milk, one teaspoon at a time.
While the doughnuts are still warm, add a handful of doughnuts to the glaze bowl and coat them gently with the glaze. Transfer them to a wire rack to dry. Allow the doughnuts to sit for at least 10 minutes to let the glaze set. Enjoy!