In a small bowl, combine warm milk and warm water, then sprinkle the active dry yeast over the top. Let it sit undisturbed for about 5 minutes until it becomes frothy.

In a large mixing bowl, combine the activated yeast mixture with sugar, softened butter, and eggs. Gradually add in the all-purpose flour, mixing until a soft dough forms.

Transfer the dough to a floured surface and knead for about 8-10 minutes, until the dough is smooth and slightly tacky.

Place the kneaded dough in an oiled bowl, covering it with a clean kitchen towel. Let it rise in a warm, draft-free location for 1 to 1.5 hours, or until it doubles in size.

While the dough rises, brown the unsalted butter in a saucepan over medium heat until it turns golden. Stir in brown sugar, ground cinnamon, and maple syrup until combined.

After the dough has risen, punch it down and roll it out on a floured surface into a 16x12 inch rectangle. Spread the filling evenly over the surface.

Starting from the longer edge, tightly roll the dough into a log. Cut the log into 12 equal pieces and place them in a greased baking dish.

Preheat your oven to 350°F (175°C) while the rolls finish rising. Bake for 25-30 minutes until golden brown on top.

In a saucepan, brown unsalted butter again. In a mixing bowl, whisk together powdered sugar, heavy cream, maple syrup, and vanilla extract. Mix in the browned butter.

Once the rolls have cooled slightly, drizzle the warm maple frosting over the top.