Warm the water and milk together until about 105–115°F (warm to the touch). In a small bowl, combine warm water, warm milk and 1 tsp of the granulated sugar; sprinkle the instant yeast over the top and let stand 5–7 minutes until foamy.

Mix dry ingredients

In a large bowl or the bowl of a stand mixer, whisk together the flour, remaining granulated sugar (about 2 tbsp), and salt.

Combine to form dough

Add the foamed yeast mixture, softened butter, and whole egg to the dry ingredients. Mix with the dough hook on low speed until ingredients come together, then increase to medium and knead 6–8 minutes until a smooth, slightly tacky, elastic dough forms. If kneading by hand, turn dough onto a lightly floured surface and knead 8–10 minutes, adding a teaspoon of flour at a time only if too sticky.

Bulk rise

Shape the dough into a ball and place in a lightly oiled bowl, turning once to coat. Cover with plastic wrap or a damp towel and let rise in a warm draft-free area until doubled in size, about 60–75 minutes depending on temperature.

Divide and shape

Punch down the risen dough gently to release gas. Turn onto a lightly floured surface and divide into 10 equal pieces (about 55–65 g each). Shape each piece into a smooth ball by tucking edges under and rolling on the counter.

Final proof

Place the dough balls on a parchment-lined baking tray or in individual muffin cups, leaving about 1–1.5 inches between. Cover loosely with plastic wrap or a lightly greased piece of plastic and let proof until puffy and nearly doubled, about 30–45 minutes.

Coffee Flavored Filling / Topping

Prepare coffee butter filling

While dough proofs, make the coffee filling. In a medium bowl, whisk together the softened butter (6 tbsp) and light brown sugar until smooth and slightly airy. Dissolve the instant espresso powder into 1 tablespoon hot brewed coffee, then stir into the butter mixture along with the vanilla extract. Add the 3 tbsp flour and mix until combined — the flour helps give body to the spread so it stays on the bun.

Assemble topping pattern

Once buns have proofed, using the back of a spoon or a small offset spatula, spread about 1 tablespoon of the coffee butter mixture over the top of each bun, leaving a small rim at the edge. For an authentic look, use a fork to create a lattice or crosshatch pattern in the spread, or pipe lines if using a pastry bag.

Optional glaze and sprinkle

If using, whisk the egg yolk with 2 tablespoons milk then brush the exposed edges of the bun (not the coffee butter topping) very lightly to give shine to the roll sides. Sprinkle coarse sugar over the tops for crunch.

Bake

Preheat the oven to 350°F (175°C). Bake the assembled buns on the center rack for 14–18 minutes, rotating the tray once halfway, until the buns are golden brown and the coffee topping looks set but not burned. If topping browns too quickly, tent loosely with foil for the last few minutes.

Finish

Remove from oven and let cool on the tray for 5 minutes, then transfer to a wire rack to cool another 10–15 minutes. The topping will firm slightly as it cools. Serve warm or at room temperature.