In a small bowl combine warm water (about 105–115°F / 40–46°C) and 1 tsp sugar from the measured amount; sprinkle instant yeast over top and let sit 5–8 minutes until foamy. If it doesn't foam, replace the yeast.

Mix dry ingredients

In the bowl of a stand mixer fitted with a dough hook (or in a large mixing bowl), combine 3.5 cups all-purpose flour, 0.25 cup granulated sugar, and 1 tsp salt. Whisk briefly to distribute.

Add wet ingredients

Add the foamed yeast and warm water to the dry mixture, plus 1 cup lukewarm whole milk, 4 tbs softened butter, and 1 large egg. Mix on low speed until dough comes together (about 2–3 minutes).

Knead

Increase speed to medium-low and knead the dough for 6–8 minutes until smooth, elastic, and slightly tacky. If dough is too sticky, add up to 2 tbs more flour total; if too dry, add 1–2 tbs milk.

First rise

Form the dough into a ball, lightly oil a bowl, place dough inside, cover with plastic wrap or a damp towel, and let rise in a warm draft-free spot until doubled in size, about 1 to 1¼ hours.

Coffee-sugar filling

Prepare filling

While dough rises, make the filling: in a medium bowl cream together 6 tbs softened butter and 0.75 cup packed dark brown sugar until smooth. Stir in 2 tsp instant espresso powder, 1 tsp vanilla extract, 0.25 tsp ground cinnamon (if using), and 1/8 tsp salt until evenly combined. The mixture should be spreadable but not runny. If too stiff, soften slightly in short bursts in the microwave (5–7 seconds).

Assembly and second rise

Roll out dough

Turn the risen dough out onto a lightly floured surface and gently deflate. Roll into a rectangle roughly 12 x 18 inches (30 x 45 cm), with the longer side closest to you; roll to about 1/8–1/4 inch thickness.

Spread filling

Evenly spread the coffee-sugar butter across the rolled dough, leaving a 1/2-inch border along the top edge. For a deeper coffee flavor, dissolve the 0.5 tsp optional instant espresso in 1 tsp hot water and brush lightly over the dough before spreading filling.

Form log and cut

Starting from the long edge nearest you, tightly roll the dough into a log. Pinch seam to seal. Using a sharp knife or dental floss, cut the log into 12 equal pieces (about 1 to 1 1/4 inch slices).

Arrange and second rise

Place buns cut-side up into two greased 9-inch round cake pans (6 per pan) or a 9x13 pan, spacing slightly apart for expansion. Cover loosely with lightly greased plastic wrap or a damp towel and let rise for 30–45 minutes until puffy and nearly doubled.

Baking and finishing

Preheat oven & egg wash

Preheat oven to 350°F (175°C). In a small bowl whisk together 1 egg and 1 tbs milk for egg wash.

Brush and sugar

Gently brush the tops of the risen buns with the egg wash, avoiding excess pooling between layers. Sprinkle 2 tbs coarse sugar evenly over the tops for crunch.

Bake

Bake in the center of the preheated oven for 18–22 minutes, or until the buns are golden brown on top and reach an internal temperature of about 190°F (88°C). If the tops are browning too quickly, tent loosely with foil for the last 5–7 minutes.

Cool

Remove buns from oven and allow to cool in pans for 10 minutes, then transfer to a wire rack. Serve warm or at room temperature. Buns will keep in an airtight container at room temperature for 2 days or refrigerated for up to 5 days; reheat gently before serving.