Pour the warm milk into a small bowl. Add the yeast and about 1 teaspoon of the measured sugar. Stir gently and let the mixture stand for 5–10 minutes, until slightly foamy. The milk should be warm but not hot; excessive heat can damage the yeast.

In the bowl of a stand mixer fitted with the dough hook, combine the flour, remaining sugar, and salt. Mix briefly to distribute the ingredients evenly.

Add the eggs and activated yeast mixture to the dry ingredients. Mix on low speed until a rough, sticky dough begins to form. Increase to medium-low speed and knead for about 5–7 minutes, until the dough starts becoming smoother and more elastic.

With the mixer running on low speed, add the softened butter a few pieces at a time. Wait until each addition is mostly incorporated before adding more. At first, the dough may look greasy or separated; continue kneading and it will gradually come back together.

Once all the butter has been incorporated, knead on medium-low speed for another 8–12 minutes. The finished dough should be smooth, glossy, stretchy, and slightly tacky. To check the dough, gently stretch a small piece between your fingers; it should become thin enough to let light through without tearing immediately — this is known as the windowpane test. Avoid adding excessive flour simply because brioche dough feels softer and stickier than ordinary bread dough.

Shape the dough gently into a ball and place it in a lightly greased bowl. Cover and let rise in a warm, draft-free place for approximately 1½ hours, or until noticeably expanded and close to doubled in size.

Gently press the risen dough to release large air bubbles. Divide it into 3 or 4 equal portions and shape each piece into a smooth ball. Arrange the balls side by side in a greased 9 × 5-inch loaf pan. This creates the beautiful rounded top commonly associated with brioche loaves.

Cover the loaf loosely and let it rise for another 45–60 minutes, or until the dough looks very puffy and rises close to the top of the pan. Avoid letting it overproof, as this can cause the brioche to collapse during baking.

Preheat the oven to 350°F (175°C). Whisk together the egg and 1 tablespoon of milk. Very gently brush the mixture over the risen brioche, being careful not to deflate the dough.

Bake for 30–35 minutes, until the brioche is deeply golden brown on top. If the top browns too quickly, loosely cover it with aluminum foil during the final 10 minutes. The center should reach approximately 190°F (88°C) when checked with an instant-read thermometer.

Let the brioche cool in the pan for about 10 minutes, then carefully transfer it to a wire rack. Allow it to cool for at least 30 minutes before slicing so the delicate interior has time to set.