Warm the milk to about 105–115°F (warm to the wrist). Stir the granulated sugar into the milk, sprinkle the instant yeast over the surface, and let sit 5–7 minutes until foamy.

Combine dry ingredients

In a large mixing bowl or the bowl of a stand mixer fitted with a dough hook, whisk together the all-purpose flour and salt.

Make dough

Add the foamy milk-yeast mixture, softened (melted then cooled) unsalted butter (3 tbs), and the beaten large egg to the flour. Mix on low speed until a shaggy dough forms, then increase to medium and knead about 6–8 minutes until the dough is smooth, elastic, and slightly tacky. If dough is too sticky, add up to 2 tablespoons flour; if too dry, add 1 tablespoon warm milk.

First rise

Lightly oil a clean bowl, place the dough inside, cover with plastic wrap or a damp towel, and let rise in a warm draft-free spot until doubled in size, about 60–90 minutes depending on room temperature.

Sweet filling

Prepare filling

While the dough rises, combine the brown sugar, ground cinnamon, softened butter (3 tbs), vanilla extract, and optional chopped nuts and raisins in a bowl. Mix until evenly distributed and spreadable. Set aside at room temperature.

Assemble and bake

Roll out dough

After the dough has doubled, gently punch it down and turn it onto a lightly floured surface. Roll into a rectangle about 12 x 16 inches (roughly 1/4-inch thick).

Spread filling

Evenly spread the prepared sweet filling over the dough, leaving a 1/2-inch bare border along one long edge to help seal the roll.

Form roll

Starting from the long side opposite the bare edge, tightly roll the dough into a log. Pinch the seam to seal, then tuck the ends under to create a neat log.

Cut rolls

Using a sharp knife or unflavored dental floss, slice the log into 10–12 equal rolls (about 1 to 1 1/4 inches wide). Place the rolls cut-side up in a lightly buttered 9×13-inch baking pan or two smaller pans, spacing them slightly apart.

Second rise

Cover the pan(s) with a clean towel or plastic wrap and let the rolls rise in a warm place until puffy and nearly doubled, about 30–45 minutes.

Preheat and egg wash

Preheat the oven to 350°F (175°C). Brush the tops of the risen rolls lightly with the beaten egg wash and sprinkle with coarse sugar if using.

Bake

Bake in the preheated oven for 20–25 minutes, until the rolls are golden brown on top and cooked through (internal temp ~190°F if measured). If tops brown too quickly, tent loosely with foil for the last 5–10 minutes.

Glaze / finishing

Make glaze

While the rolls bake, whisk together the powdered sugar, milk (start with 2 tbs), and vanilla extract until smooth. If too thick, add additional milk 1/2 teaspoon at a time until desired drizzling consistency is reached.

Glaze and serve

When rolls come out of the oven, let them cool in the pan 8–10 minutes. Either drizzle the glaze over warm rolls or spread it for fuller coverage. Serve warm. Leftovers keep in an airtight container at room temperature up to 2 days or refrigerated for up to 4 days; reheat gently before serving.