Preheat your oven to 350°F for about 30 seconds, then turn it off and leave the oven light on. This creates a warm place for the dough to rise later.
In a stand mixer bowl, combine the warm water, vegetable oil, sugar, egg, yeast, and salt.
Add 3 ¼ cups of flour and knead with the dough hook for 5 minutes until it no longer sticks to the sides of the bowl. If the dough is still sticky, gradually add up to ¼ cup more.
Divide the dough into 8 pieces and roll them into balls. Place on a large parchment-lined baking sheet, spacing them about 3 inches apart. Gently flatten each ball slightly.
Cover the baking sheet and place it in the warm oven. Let the buns rise for 20-30 minutes.
Remove the buns from the oven and preheat the oven to 425°F.
Brush with egg wash and bake for 8-12 minutes until golden brown.