Add the flour, sugar, salt, eggs and yolk, milk, and sourdough starter to the bowl of a stand mixer. Mix on low speed until a rough dough forms.
Increase the speed slightly and knead until the dough begins to smooth out and pull together.
With the mixer running on low, add the very soft butter a little at a time, allowing it to fully absorb before adding more.
After adding the butter, mix for 10 to 15 minutes on low to medium speed, or until the dough becomes smooth, glossy, and elastic, and releases from the sides of the bowl. This step takes time and is key to proper brioche structure.
Transfer the dough to a lightly greased bowl, cover, and allow it to bulk ferment at room temperature until noticeably puffy and lighter in texture. During bulk fermentation, the dough should rise about 60 to 80 percent in volume and feel noticeably lighter, rather than fully doubling.
Gently turn out the dough, shape it, and place it into a standard 9×5-inch loaf pan.