Heat milk and 1/3 cup butter in a microwave-safe container in the 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.

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.

Mix together 3 cups flour, sugar, yeast, and salt in a large bowl until well combined. Add to milk mixture and beat on low speed using the dough hook, scraping down the sides of the bowl using a rubber spatula as needed.

Add remaining 1 cup 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.

Pour oil over dough and turn to coat. Cover and let rest at room temperature for 10 minutes.

Turn dough out onto a lightly floured surface and use your hands to shape into a small rectangle. Roll dough out into a 1/4-inch-thick rectangle.

Mix together brown sugar and cinnamon in a small bowl until well combined. 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.

Place rolls, cut-side down and almost touching, into a greased 9x13-inch pan. Cover loosely with plastic wrap and let rise at room temperature until doubled in size, 30 to 40 minutes.

Preheat the oven to 400 degrees F (200 degrees C).