In a large bowl, combine the butter, honey, oats, and salt. Pour in the hot water and stir until the butter is dissolved, 1-2 minutes. Allow to cool to no less than 110°F, 5-6 minutes.
Stir in the milk with the beaten egg and add 2 cups of the flour and yeast. Stir to incorporate, creating a sticky dough. Add more flour, 1 cup at a time, incorporating well after each addition. Stir and eventually knead the dough. Do not add more flour, but knead the dough until it is slightly tacky and clears the side of the bowl, 3-5 minutes. Form the dough into a smooth ball.
Drizzle the sides of the bowl and the dough with 1 tbsp neutral oil. Cover and allow the dough to rise in a warm place until doubled, 40-60 minutes.
Once doubled, punch down the dough and knead into a ball. Separate the dough into 15 even-sized dinner rolls and place in a buttered 9x13 baking dish. Cover and let rise until doubled, 35-50 minutes.
Preheat oven to 350°F. Once dinner rolls have doubled, place in the preheated oven and bake until the tops are golden, 20-25 minutes. The center of the bread should reach 190°F. Remove from the oven and allow to cool 20 minutes before pulling apart and serving.