Proof yeast in warm coffee with 1 tsp sugar for 5 to 10 minutes until foamy. Whisk 2 3/4 cups flour, remaining sugar, and salt in a large bowl. Add yeast mixture, melted butter, and egg; stir until a shaggy dough forms. Knead 8 to 10 minutes, adding remaining flour as needed. Place in a greased bowl, cover, and let rise until doubled, about 1 hour.

Punch down dough and roll into a 16x12-inch rectangle. Spread 3 tbsp soft butter over the surface, leaving a border at the top edge. Mix brown sugar, cinnamon, and 2 to 3 tsp espresso powder; sprinkle over butter and press lightly. Roll tightly from the long side, pinch seam to seal.

Cut log into 12 even slices and place cut side up in a greased 9x13-inch pan. Cover and let rise until puffy, 45 to 60 minutes. Preheat oven to 350°F (175°C) toward the end of the rise.

Bake for 20 to 25 minutes until golden and the center roll springs back. Let cool in the pan for 10 minutes.

Meanwhile, beat cream cheese and 1/4 cup butter until fluffy. Add powdered sugar, 2 tsp espresso powder, vanilla, and salt; beat until light and spreadable.

Spread frosting over warm rolls and serve right from the pan.