In the bowl of a stand mixer fitted with the paddle attachment, combine 3 cups (375 grams) flour, sugar, salt, and yeast by hand. Add Flour Paste.
In a medium saucepan, heat ½ cup (120 grams) water, ⅓ cup (80 grams) milk, and butter over medium heat until an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add warm milk mixture to flour mixture, and beat at low speed until combined. Beat in 1 egg (50 grams).
Switch to the dough hook attachment. Beat at low speed until a soft, somewhat sticky dough forms, 8 to 9 minutes, stopping to scrape sides of bowl and dough hook. Add up to remaining ¼ cup (31 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky. Turn out dough onto a lightly floured surface, and shape into a smooth round.
Lightly oil a large bowl. 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, 45 minutes to 1 hour.
Line a baking sheet with parchment paper.
Punch down dough, and let stand for 5 minutes. Turn out onto a lightly floured surface. Divide dough into 8 portions (about 97 grams each). Press 1 portion into a 5×4-inch rectangle. (Keep remaining dough covered to prevent it from drying out.) Place rectangle widthwise in front of you. Starting with bottom 5-inch edge, fold dough, about ½ inch at a time, pressing to seal each time before folding again. Repeat until you have a tight cylinder. Place seam side up, and fold ends in. Place seam side down, and shape to 5 inches long and even thickness. Place on prepared pan, and press gently to flatten top. Repeat with a second portion of dough, and place about ¼ inch from first bun; press flat. Repeat with remaining dough, making 2 rows of 4 buns. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, 30 to 45 minutes.
Preheat oven to 350°F (180°C).
In a small bowl, whisk together remaining 1 egg (50 grams) and remaining 1 tablespoon (15 grams) milk. Brush buns with egg wash.
Bake until golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), 18 to 20 minutes. Let cool on pan for 10 minutes. Remove from pan, and let cool completely on a wire rack before slicing.