In the bowl of a stand mixer fitted with the paddle attachment, whisk together warm milk, 1 tablespoon (12 grams) sugar, and yeast. Let stand until mixture is foamy, about 10 minutes.

Add Milk Roux, flour, melted butter, 1 egg (50 grams), salt, vanilla, and remaining 4½ tablespoons (54 grams) sugar to yeast mixture; beat at low speed just until combined, stopping to scrape sides of bowl. Switch to the dough hook attachment; beat at low speed for 3 minutes. Increase mixer speed to medium-low, and beat until a soft, smooth dough forms, 6 to 8 minutes, stopping to scrape sides of bowl. (Dough should pass the windowpane test; see Note.) Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, about 1 hour.

Line 2 baking sheets with parchment paper.

Punch down dough; let stand for 5 minutes. Divide dough into 10 equal portions (about 77 grams each). Roll each portion into a ball, and place on prepared pans.

Divide Pineapple Buns Topping into 10 portions (about 27 grams each); roll into balls. Place 1 ball between 2 sheets of plastic wrap; roll or pat into a 4-inch circle. Gently wrap circle over top of 1 bun, covering as much dough as possible. Repeat with remaining Pineapple Buns Topping and buns. Using a sharp knife, gently cut a crosshatch pattern into top of each bun through topping. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 40 minutes to 1 hour.

Preheat oven to 350°F (180°C).

In a small bowl, whisk together room temperature milk and remaining 1 egg (50 grams). Thoroughly brush top of buns with egg wash, being sure to get between rows of topping.

Bake until golden brown and an instant-read thermometer inserted in center registers at least 190°F (88°C), about 15 minutes. Let cool on pans for 10 minutes before serving.