In a large bowl, combine warm milk and yeast. Let it sit until the yeast dissolves for 3-5 minutes.
In a microwaveable bowl, melt the butter in the microwave for 1-1½ minutes. After a minute, stir until the butter melts. Let the butter cool to about 105-110℉ (40-43℃).
Pour the warm (not hot) butter into the yeast mixture, and then add vanilla extract, sugar, eggs, orange juice, orange zest, and salt, and mix well.
Add the flour, one cup at a time, mixing with your hands or in a stand mixer. The dough will be sticky, but if it's too sticky, you can add a tad more flour. Less flour is preferable for moist cinnamon rolls.
Cover the dough loosely with a clean cloth and let it rise in a warm, draft-free place for an hour or two or until doubled in size.
Combine the sugar, cinnamon, and orange zest. Set aside until ready to use.
Punch the dough down, then lightly flour your work surface and roll the dough into a rectangle about an inch thick. Brush with melted butter, leaving approximately 1 inch around the border. Sprinkle with the sugar mixture and pat it gently on the surface.
Starting with the end nearest you, roll the dough up nice and snug like a jelly roll around the filling, finishing the roll with the seam side down. Don't forget to pinch the dough to seal it so it doesn't come apart while baking.
Slice the roll into four equal sections, then slice each section into three equal parts for a total of twelve rolls.
Oil the bottom of the baking pan with butter or cooking spray. Place orange cinnamon roll slices close together in the pan and let them rise until they double in size, about 45 minutes.
Bake the orange cinnamon rolls for 25-30 minutes or until lightly golden brown (be careful not to overbake). Remove them from the oven and let them cool for about 5 minutes.