Prepare the Sponge: In a mixing bowl, combine the sourdough discard, warm milk, sugar, and flour. Stir until smooth. Cover with a towel and let it sit in a warm place for 1 hour. The sponge should become bubbly and slightly puffy.

Mix the Dough: In a stand mixer fitted with a dough hook, combine the flour, sugar, and salt. Add the sponge and mix on low speed. Gradually add the eggs, one at a time, ensuring each is fully incorporated before adding the next.

Knead the Dough: Continue to mix on medium speed, gradually adding the softened butter, one tablespoon at a time. Knead the dough for about 10-15 minutes, or until it’s smooth and elastic. The dough should be soft but not sticky.

First Rise: Place the dough in a lightly greased bowl, cover with plastic wrap, and let it rise in a warm place for 1-2 hours, or until doubled in size.

Shape the Dough: Once risen, punch down the dough to release the air. Turn it out onto a lightly floured surface and divide it into three equal pieces. Shape each piece into a ball and place them side by side in a greased loaf pan.

Second Rise: Cover the loaf pan with a towel and let it rise for another 1-2 hours, or until the dough has risen above the edges of the pan.

Preheat the Oven: While the dough is rising, preheat your oven to 375°F (190°C).

Egg Wash: In a small bowl, whisk together the egg yolk and milk. Gently brush the top of the risen dough with the egg wash, which will give the brioche a beautiful golden-brown color when baked.

Bake: Bake the brioche in the preheated oven for 25-30 minutes, or until the top is golden brown and the bread sounds hollow when tapped. If the top is browning too quickly, cover it loosely with aluminum foil.

Cool and Serve: Let the brioche cool in the pan for 10 minutes before transferring it to a wire rack to cool completely. Slice and enjoy!