Mix milk, starter, honey, egg, oil, and salt in the bowl of your stand mixer and pulse until combined.
Add 3 cups of the flour and mix well.
Slowly add the remaining cup of flour. Mix until combined.
Knead in the mixer for 10-15 minutes or until glossy and smooth. It should pass the window pane test.
Transfer dough to greased mixing bowl. Cover and allow to ferment for 4-12 hours. Dough should be puffy with noticeable bubbles on top of the dough.
Transfer dough to lightly floured work surface. Divide into 8 equal parts.
Roll each portion into a rectangle. Roll from side to side to make hoagie roll shape.
Place on parchment lined baking sheet. Cover and allow to rise for 1-2 hours or until noticeably puffy and airy.
Preheat oven to 375 degrees F.
Brush with egg wash.
Bake for 18-20 minutes until golden brown.