Warm the milk to about 110°F (43°C). In a large bowl or the bowl of a stand mixer, stir together the milk, yeast and 1 tablespoon of the sugar. Leave for 5–10 minutes, until a foamy layer forms. If the mixture stays flat, check the yeast and milk temperature before continuing.
Stir in the melted, slightly cooled butter, remaining 1 tablespoon sugar, salt and room-temperature egg. Add 3 cups of the flour gradually, mixing with a dough hook on low speed or a sturdy spoon. If the dough is too wet to knead, add the remaining ½ cup flour a tablespoon at a time, only as needed. The dough should be soft and slightly tacky.
Knead for 5–7 minutes with a dough hook, or about 8–10 minutes by hand, until smooth and stretchy. Work in the 1½ cups grated cheddar until evenly distributed. Shape into a ball and place in a lightly greased bowl.
Cover the bowl with a lid or lightly greased plastic wrap. Leave in a warm, draft-free spot for 60–90 minutes, until roughly doubled in size. Use the size of the dough as your guide; a cool room may need longer.
Line a 13-by-18-inch half-sheet pan with parchment. Gently press down the risen dough and divide into 12 equal pieces. Pull the edges of each piece underneath, pinch the seam closed and round into a ball. Arrange seam-side down on the pan with room between the buns.
Cover loosely with lightly greased plastic wrap and let rise for 30–45 minutes, until puffy. During this rise, preheat the oven to 375°F (190°C) with a rack in the middle.
Whisk the egg-wash egg with 1 tablespoon milk or water. Uncover the buns and brush the tops gently with a thin layer of egg wash. If using the optional ¼ cup cheddar, scatter it evenly over the twelve buns.
Bake for about 15–18 minutes, until the tops are golden and the sides are lightly browned. The center of a middle bun should reach about 190°F (88°C); give it a few more minutes if needed. If the tops brown too quickly, cover loosely with foil for the remaining baking time.
Move the buns to a wire rack. Let cool for about 10 minutes before serving warm.