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 the surface becomes foamy and bubbly, indicating 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 electric mixer fitted with a dough hook 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 clean kitchen towel and set in a warm, draft-free spot for 60 minutes, or until the dough has doubled in bulk.
05 - While the dough rises, preheat the oven to 375°F (190°C). Line a 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 knead in 1 1/4 cups (125 g) of the grated sharp Cheddar until evenly distributed throughout the dough.
07 - Divide the cheese-studded dough into 8 equal portions. Shape each piece into a smooth, round bun by pulling the edges underneath and rolling against the counter. Arrange the buns on the prepared baking sheet, spacing them about 2 inches apart.
08 - Sprinkle the remaining 1/2 cup (50 g) grated Cheddar and optional sesame or poppy seeds over the tops of the buns. Cover loosely with a towel and let rise for 20 to 30 minutes until puffy and nearly doubled.
09 - Lightly brush the tops of the buns with milk using a pastry brush. Bake on the center rack for 18 to 20 minutes, until the tops are deep golden brown and the cheese is melted and bubbly.
10 - Remove the buns from the oven and allow them to cool on the baking sheet for 5 minutes. Transfer to a wire rack. Serve warm or at room temperature.