In a small bowl, sprinkle 2 teaspoons of active dry yeast over the warm milk (110°F) with a pinch of the granulated sugar. Let it sit for 5–10 minutes until foamy and fragrant.

In a mixing bowl, whisk together the remaining granulated sugar, egg, and 1 teaspoon vanilla. Stir in the foamy yeast mixture, melted butter, and gradually add 3 cups all-purpose flour mixed with 1 teaspoon salt. Mix until a soft dough forms; it should be slightly tacky but manageable.

Turn the dough onto a lightly floured surface and knead gently for 6–8 minutes until smooth and elastic. Place in a lightly oiled bowl, cover with a clean towel or plastic wrap, and let it rise in a warm place until doubled, about 1 to 1½ hours.

Once risen, roll the dough to about 1/2-inch thickness. Use a round cutter to cut circles (about 3 inches) and place them on a parchment-lined baking sheet, leaving space for a second rise. Cover and let rise for 30–45 minutes until puffy.

In a deep skillet or heavy pot, heat 4 cups of oil (vegetable or canola) to 350°F (175°C). Maintain the temperature—too cool and donuts soak oil; too hot and they brown before cooking through.

Fry donuts in small batches, 1–2 minutes per side, until a golden, even brown. Use a slotted spoon to turn and remove, then drain on a wire rack set over a baking sheet. Allow donuts to cool until warm but not hot.

Whisk together 1 cup heavy cream, 1/2 cup powdered sugar, 2 tablespoons instant coffee granules, and 1 teaspoon vanilla extract. Whip until medium-stiff peaks form—silky, billowy, and holding shape. Taste and adjust coffee intensity if desired. Transfer to a piping bag fitted with a plain tip.

In a bowl, sift 1 cup powdered sugar and whisk in 1/4 cup brewed coffee (cooled) until a smooth, glossy glaze forms. It should be thick enough to coat but thin enough to drip just slightly.

Using a small tip, pipe coffee cream into the side of each donut until gently full. Dip the tops into the coffee glaze, letting excess drip back into the bowl. Set on the cooling rack to allow the glaze to set to a soft sheen.

For a textural contrast, dust a light veil of powdered sugar or sprinkle finely crushed coffee beans over the glaze. Serve warm for the most tender bite.