Make the Basic Brioche Dough: In the bowl of a stand mixer fitted with the paddle attachment, combine warm milk, sugar, and yeast. Let stand until mixture is foamy, about 10 minutes.
With mixer on medium-low speed, add 1 ½ cups (188 grams) flour and eggs, beating until smooth, 2 to 3 minutes. Cover and let stand for 30 to 45 minutes.
Switch to the dough hook attachment. Add salt and remaining 1 ¾ cups (18 grams) flour, and beat at medium speed until a smooth and elastic dough forms and pulls away from sides of bowl, 8 to 10 minutes.
With mixer on medium speed, add butter, 1 tablespoon (14 grams) at a time, letting each piece incorporate before adding the next.
Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75 F) until doubled in size, 1 ½ to 2 ½ hours.
On a lightly floured surface, turn out dough, and fold a few times to knock out a bit of air. Return dough to greased bowl; cover and refrigerate for at least 8 hours or overnight.
Make the chocolate brioche: Butter and flour a 9x5-inch loaf pan. On a lightly floured surface, turn out Basic Brioche Dough. Punch down dough, and sprinkle chocolate onto dough. Knead chocolate into dough briefly, just until chocolate is incorporated.
Shape dough into a loaf. Place dough, seam side down, in prepared pan. Cover and let rise in a warm, draft-free place (75 F) until dough is puffed, 30 to 40 minutes.
Preheat oven to 400 F.
Brush top of loaf with egg wash, and bake for 15 minutes.
Reduce oven temperature to 350 F, and bake until golden brown and an instant-read thermometer inserted in center registers 190 F, 35 to 40 minutes more.