Preheat oven to 425 degrees F (218 degrees C).
Into the bowl of a stand mixer fitted with a paddle attachment, add Water (1 cup), Water (2 tablespoon), Granulated Sugar (1/4 cup), Salt (1 teaspoon), Active Dry Yeast (2 tablespoon), and Vegetable Oil (1/3 cup). Mix until smooth, then let sit for 5 minutes.
Mix briefly to remove the froth from the top of the yeast mixture.
Add 2 cups of All-Purpose Flour (2 cup) and Egg (1). Mix.
Add 1/2 cup of All-Purpose Flour (1/2 cup). Mix.
Change paddle attachment to a dough hook and add remaining All-Purpose Flour (1/2 cup). Mix on low until dough comes together, about 5-6 minutes.
Transfer dough to a lightly floured surface. Knead until dough is no longer sticky.
Split dough into 8 balls, weighing about 4.25 ounces each. Roll them out to 7 inches in length and 2 inches in width.
Let dough sit for 10-15 minutes on a baking sheet or pan, then gently stretch each bun to make it about 10 inches long.
Bake for 12-14 minutes, or until golden brown.
Transfer buns to a cooling rack before serving.