Step 1
Heat milk and 1/3 cup butter together in a microwave-safe container in a microwave on high power for 30 seconds. Stir and repeat until mixture reaches 120 to 130 degrees F (49 to 54 degrees C). If too cool microwave for 15 seconds more; if too hot let cool to proper temperature.
Step 2
Whisk eggs lightly in a small bowl adding a small amount of milk mixture at a time to temper eggs until thoroughly combined. Pour mixture into the bowl of an electric stand mixer.
Step 3
Mix 3 cups flour sugar yeast and salt together in a large mixing bowl. Add to milk mixture and beat on low speed using a dough hook scraping down the sides of the bowl using a rubber spatula as needed.
Step 4
Add remaining flour slowly 1 tablespoon at a time until dough begins to pull away from the sides of the bowl. Continue to knead on low speed until dough looks smooth 7 to 10 minutes; dough will be sticky and the bottom will remain stuck to the bowl.
Step 5
Pour vegetable oil over dough and turn to coat. Cover and let rest for 10 minutes.
Step 6
Turn dough out onto a lightly floured surface and use your hands to shape into a small rectangle. Roll dough out into a rectangle that is about 1/4-inch thick.
Step 7
Mix brown sugar and cinnamon together in a small bowl. Spread 1/3 cup softened butter over dough and sprinkle with cinnamon-sugar mixture. Roll dough into a log and cut into 12 equal pieces.
Step 8
Place rolls cut-side down and almost touching into a greased 9x13-inch pan. Cover loosely with plastic wrap and let rise until doubled in size 30 to 40 minutes.
Step 9
Preheat the oven to 400 degrees F (200 degrees C).
Step 9
Bake in the preheated oven until golden 15 to 20 minutes. Remove from the oven and immediately turn out onto a tray or plate to cool; this allows the inside mixture to drip back into the rolls as they cool.