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 foamy and activated.
02 - Add the all-purpose flour, salt, softened butter, and egg to the yeast mixture. Mix thoroughly with a wooden spoon or electric mixer fitted with a dough hook until a soft, slightly tacky dough forms.
03 - Turn the dough out onto a lightly floured surface and knead by hand for 8 to 10 minutes, until the dough is smooth, elastic, and springs back when gently pressed.
04 - Place the kneaded dough in a lightly greased bowl, cover with a clean kitchen towel or plastic wrap, and let rise in a warm, draft-free spot for 60 minutes or until doubled in bulk.
05 - Preheat your oven to 375°F. Line a baking sheet with parchment paper and set aside.
06 - Gently punch down the risen dough and turn it onto a floured surface. Knead 1 1/4 cups (125 g) of the grated sharp Cheddar directly into the dough until evenly distributed.
07 - Divide the cheese-filled dough into 8 equal portions. Shape each piece into a smooth, round bun and arrange them on the prepared baking sheet, spacing at least 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 and let rise for 20 to 30 minutes until puffy.
09 - Lightly brush the tops of the risen buns with milk. Bake on the center rack for 18 to 20 minutes, until the tops are golden brown and the cheese is melted and bubbly.
10 - Remove from the oven and allow the buns to cool on the baking sheet for 5 minutes. Transfer to a wire rack. Serve warm or at room temperature.