In the bowl of a stand mixer, whisk together 2 cups (254 grams) flour, sugar, 1 tablespoon (9 grams) salt, and yeast by hand.

In a medium saucepan, heat milk, 1⁄2 cup (120 grams) water, and butter over medium heat until an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add hot milk mixture to flour mixture; using the paddle attachment, beat at medium speed until combined. Add 1 egg (50 grams), beating until combined. With mixer on low speed, gradually add 2 cups (254 grams) flour, beating just until a shaggy dough comes together and stopping to scrape sides of bowl.

Switch to the dough hook attachment. Beat at low speed until a soft, smooth, somewhat sticky dough forms, 20 to 23 minutes; add up to remaining 1⁄4 cup (32 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, 40 minutes to 1 hour.

Spray a 12-cup muffin pan with baking spray with flour.

Punch down dough, and let stand for 5 minutes. On a lightly floured surface, roll dough into an 18x12-inch rectangle, one long side facing you.

In a small bowl, stir together green onion, sesame oil, garlic, and remaining 1⁄4 teaspoon salt.

Using a pastry brush, brush melted butter onto dough. Sprinkle green onion mixture evenly onto bottom half of dough, leaving a 1⁄2-inch border along bottom long side. Fold top half down, lightly pressing to adhere. Trim 1⁄8 to 1⁄4 inch off short sides of dough; cut crosswise into 12 strips (about 1½ inches wide each).

Leaving 1⁄2 inch intact at folded end, cut 1 dough strip into 3 strands. Loosely braid strands, pinching end to adhere. Carefully tuck loose end of braid under to create a knot. Place in a prepared muffin cup. Repeat with remaining dough strips. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, 30 to 45 minutes.

Preheat oven to 375°F (190°C).

In another small bowl, whisk together remaining 1 egg (50 grams) and remaining 1 tablespoon (15 grams) water. Using a pastry brush, gently brush egg wash onto rolls; sprinkle with sesame seeds.

Bake until golden brown and an instant-read thermometer inserted in center registers at least 190°F (88°C), about 20 minutes. Let cool in pan on a wire rack for 10 minutes. Serve warm. Store in an airtight container for up to 3 days.