In a small bowl, stir together warm milk, yeast, and 1 teaspoon (4 grams) sugar until well combined. Let stand until foamy, about 10 minutes.

In the bowl of a stand mixer fitted with the paddle attachment, whisk together 2 cups (250 grams) flour, salt, and remaining ½ cup (100 grams) sugar. Add yeast mixture, and beat at low speed until combined. Add 1 egg (50 grams) and vanilla, beating until combined. With mixer on low speed, gradually add remaining cups 1⅔ cups (208 grams) flour, beating just until dough comes together and stopping to scrape sides of bowl.

Switch to the dough hook attachment. With mixer on low speed, add butter, 1 cube at a time, beating until combined after each addition. Beat until dough is elastic and smooth, 8 to 10 minutes.

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, 2 to 2½ hours. (Alternatively, cover and refrigerate overnight.)

Line 2 baking sheets with parchment paper.

Lightly punch down dough. Cover and let stand for 10 minutes. Turn out dough onto a very lightly floured surface. Divide dough into 12 portions (about 75 grams each). Using the palm of your hands, gently shape each portion of dough into a smooth ball, lightly flouring your hands as needed. Using the palm of your hand, firmly press down on dough to flatten slightly. Place 3 to 4 inches apart on prepared pans. Cover with lightly greased plastic wrap, and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 1½ to 2 hours.

Preheat oven to 375°F (190°C).

In another small bowl, whisk together 1 tablespoon (15 grams) water and remaining egg (50 grams) until well combined. Using a pastry brush, brush egg wash on top and sides of dough.

Bake until golden brown and an instant-read thermometer inserted near center registers 190°F (88°C), 8 to 10 minutes. Let cool on pans for 5 minutes. Remove from pans, and let cool completely on wire racks.