Preheat the oven to 375°F. Line sheet trays with parchment paper, set aside.
Place the softened butter, brown sugar, granulated sugar, cinnamon, and vanilla into a medium-sized bowl. Mix with an electric hand mixer until combined and smooth.
Unroll the crescent rolls and lay them on a clean work surface.
Spread a scant tablespoon of the filling over the dough.
Roll into a crescent shape and tuck the pointy ends under. Place on a prepared sheet tray about 3 inches apart. Repeat with the remaining rolls.
Bake for 11-13 minutes until golden brown on top. Some of the filling will escape while baking.
While the rolls are baking, make the glaze by whisking the powdered sugar, 3 tablespoons of the milk, and the vanilla into a shallow bowl. If it seems too thick, add up to another tablespoon of milk to reach the desired consistency. It should be thick yet pourable.
Allow the rolls to cool slightly. Dip the tops of the rolls one by one into the glaze, allowing any excess to drip off.
Place them glaze side up onto a wire rack over a sheet tray to catch any drips. Repeat with the remaining rolls.
Wait 15-30 minutes to allow the glaze to harden and serve immediately.