01 - In a large mixing bowl, combine the warm water, warm milk, sugar, and active dry yeast. Stir gently and let the mixture rest for 5 minutes until it becomes foamy and bubbly, confirming the yeast is active.
02 - Add the all-purpose flour, salt, softened butter, and egg to the yeast mixture. Stir with a wooden spoon or mix with an electric mixer on low speed until a soft, slightly tacky dough forms and pulls away from the sides of the bowl.
03 - Turn the dough out onto a lightly floured surface and knead by hand for 8 to 10 minutes, until the dough becomes smooth, elastic, and springs back when gently pressed with a fingertip.
04 - Place the kneaded dough in a large greased bowl, turning once to coat all sides. Cover tightly with plastic wrap or a damp kitchen towel. Set in a warm, draft-free spot and let rise for 60 minutes, or until the dough has doubled in bulk.
05 - Preheat the oven to 375°F (190°C). Line a large baking sheet with parchment paper and set aside.
06 - Gently punch down the risen dough to release air bubbles. Turn it out onto a floured surface and gradually knead in 1 1/4 cups (125 g) of the grated sharp Cheddar until evenly distributed throughout the dough.
07 - Divide the cheese-filled dough into 8 equal portions. Shape each piece into a smooth, round bun by pulling the edges toward the center and rolling against the counter. Arrange the shaped buns on the prepared baking sheet, spacing them about 2 inches apart.
08 - Sprinkle the remaining 1/2 cup (50 g) grated Cheddar evenly over the tops of the buns. Add sesame seeds or poppy seeds if desired. Cover loosely with a towel and let rise in a warm spot for 20 to 30 minutes until puffed and nearly doubled.
09 - Lightly brush the tops of the risen buns with milk using a pastry brush for a golden finish. Bake on the center rack for 18 to 20 minutes, until the buns are deep golden brown and the cheese on top is melted and bubbly.
10 - Remove the buns from the oven and let them cool on the baking sheet for 5 minutes. Transfer to a wire rack to cool slightly. Serve warm or at room temperature.