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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. 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 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. 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. Pour vegetable oil over dough and turn to coat. Cover and let rest 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 rectangle that is about 1/4-inch thick. 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. 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. Preheat the oven to 400 degrees F (200 degrees C). 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.