In the bowl of a stand mixer fitted with the paddle attachment, combine warm milk, sugar, and yeast. Let stand for 10 minutes.
Add 1½ cups (188 grams) flour and 5 eggs (250 grams) to yeast mixture, and beat at medium-low speed until smooth, 2 to 3 minutes. Cover and let stand until small bubbles form around edges of mixture, 30 to 45 minutes.
Switch to the dough hook attachment. Add salt and remaining 1¾ cups (218 grams) flour. Beat at medium speed until dough pulls away from sides of bowl and is smooth and elastic, 8 to 10 minutes.
With mixer on medium speed, add butter, 1 tablespoon (14 grams) at a time, beating until combined after each addition. Add cheese and rosemary. (See Note.)
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/24°C) until doubled in size, 1½ to 2½ hours.
Turn out dough onto a lightly floured surface, 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 (up to 16 hours).
Lightly spray 2 (9x5-inch) loaf pans with cooking spray.
Divide dough into 6 portions (about 200 grams each). On a lightly floured surface, roll each portion into a 12-inch-long rope. Braid 3 ropes of dough very tightly. Place in 1 prepared pan, tucking ends under. Repeat procedure with remaining 3 ropes of dough and remaining prepared pan. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 1 to 1½ hours.
Preheat oven to 350°F (180°C).
In a small bowl, whisk together 1 tablespoon water and remaining 1 egg (50 grams). Brush egg wash over dough.
Bake for 30 minutes. Cover with foil, and bake until an instant-read thermometer inserted in center registers 190°F (88°C), 15 to 20 minutes more. Serve warm.
Notes
Note: If cheese and rosemary don’t incorporate well in the mixer, turn out onto a lightly floured surface and knead by hand until everything is evenly distributed.
Recipe by Bake from Scratch at https://bakefromscratch.com/cheese-rosemary-brioche/