Activate Yeast: In the bowl of a stand mixer, combine warm milk, 1 tablespoon of sugar, and yeast. Let sit for 5-10 minutes until frothy.
Mix Wet Ingredients: Add the melted butter, the remaining 1 tablespoon of sugar, salt, and the egg. Whisk briefly.
Add Flour: Attach the dough hook. Gradually add 3 cups of flour on low speed. If the dough is too wet, add the remaining half cup, a tablespoon at a time, until the dough clears the sides of the bowl.
Knead & Add Cheese: Knead for 5-7 minutes until smooth and elastic. Gently knead in the grated cheddar cheese until distributed evenly.
First Rise: Form into a ball, place in a greased bowl, cover, and let rise in a warm place for 1 to 1.5 hours until doubled in size.
Shape: Punch down the dough and divide into 12 equal pieces. Roll into smooth balls and place on a parchment-lined baking sheet.
Second Rise: Cover loosely and let rise for 30-45 minutes until puffy.
Bake: Preheat oven to 375°F (190°C). Brush the buns with egg wash and top with a little extra cheese. Bake for 15-18 minutes until deeply golden brown.
Cool: Let cool on a wire rack before serving.