Sauté the Beef & Veggies: In a large skillet, heat the vegetable oil on high. Sauté the onions until translucent. Add the ground beef and cook for 4 minutes until just browned; remove the meat to a bowl and set aside. Return the skillet to high heat. Add the potatoes and cook for 3 minutes, then add the carrots and cook 3 more minutes.
Season and Thicken the Filling: Reduce heat to medium-high. Return the beef to the skillet with vegetables. Stir in the salt, curry powder, mustard, turmeric, coriander, and brown sugar; cook for 3 minutes. Stir in the frozen peas and ¼ cup of water; cook for 5 minutes. Whisk the cornstarch and 3 Tablespoons of water together. Stir the slurry into the skillet and cook for 2 minutes until thickened. Transfer to a bowl, cool, and refrigerate for 1 hour.
Mix the Dough: Preheat the oven to its lowest setting (170°F/76°C) for 10 minutes, then turn it off. In a large bowl, whisk the bread flour, cake flour, sugar, yeast, milk powder, and salt. Place half the flour mixture into a stand mixer with the paddle attachment. On low speed, slowly pour in the milk, then add the egg.
Knead and First Proof: Switch to the dough hook, add the remaining flour mixture, and knead on low for 5 minutes. Increase to medium-low and knead for 15 minutes (25 minutes if by hand). Shape the dough into a ball, place in a lightly oiled bowl, loosely cover with plastic wrap and a clean hand towel, and place in the warmed, turned-off oven for 1 hour.
Portion and Fill: Punch down the dough and turn it out onto a floured surface. Weigh the total dough, divide that number by 20, and cut the dough into 20 equal-weight balls (approx 59g each). Roll one ball into a 5-inch circle, place 2-3 Tablespoons of chilled filling in the center, pinch the edges together, and twist to seal. Repeat with remaining dough.
Second Proof and Bake: Place buns seam-side down on parchment-lined trays, spaced 2 inches apart. Cover with a towel and place back into the warmed, turned-off oven for 30 minutes. Remove trays and preheat the oven to 350°F (176°C). Whisk the egg and 1 Tablespoon water; brush over the buns. Bake for 20 to 25 minutes until golden brown. Cool on a wire rack and serve!