Start by dissolving 1 tablespoon of instant coffee in the warm milk. Give it a good stir until the coffee completely dissolves. This usually takes about 30 seconds of stirring.
In a large mixing bowl, combine 2 cups of the flour, sugar, yeast, and salt. Add your coffee-milk mixture, melted butter and eggs. Mix with a wooden spoon until everything’s well combined.
Gradually stir in the remaining flour, about 1/2 cup at a time. Once it gets too hard to stir, turn it out onto a floured surface. The dough will still be slightly sticky – that’s exactly what you want. Knead for about 8-10 minutes until it feels smooth and elastic. If you press your finger into it, it should slowly spring back.
Place the dough in a lightly greased bowl, cover with a damp cloth, and let it rise in a warm spot for about 1 hour.
While the dough rises, mix your filling. Combine the brown sugar, cinnamon, and instant coffee granules in a bowl. The coffee should be ground pretty fine – if your granules are large, crush them up a bit.
Once risen, punch down the dough (literally punch it – it’s satisfying!) and roll it out on a floured surface into a roughly 16×20 inch rectangle.
Spread the softened butter all over the dough, leaving about a 1/2 inch border around the edges. Sprinkle your coffee-cinnamon-sugar mixture evenly over the butter, patting it down gently so it sticks.
Starting from the long side, roll the dough up tightly – like you’re rolling a sleeping bag. Keep it snug but don’t pull so hard that you stretch the dough. Pinch the seam to seal it.
Cut into 12 equal pieces using dental floss or a sharp knife (about 1.5 inches each). To use floss, slide it under the roll, cross the ends over the top, and pull – this gives you clean cuts without squishing the rolls.
Place the rolls in a greased 9×13 inch baking pan, leaving some space between each one. Cover and let rise again for about 30 minutes, or until they’re puffy and touching each other.
Preheat your oven to 375°F while they’re doing their second rise. Bake for 20-25 minutes, until they’re golden brown on top and a thermometer inserted into the center roll reads 190°F. If they’re browning too quickly, tent with foil.
For the glaze, whisk together the powdered sugar, strong coffee, cream, vanilla, and salt until smooth. It should be thick but pourable – add more coffee or cream if needed, just a teaspoon at a time.
Let the rolls cool for about 5 minutes (they should still be warm but not hot), then pour the glaze over them.