Brown 6 Tbsp unsalted butter in a small saucepan over medium heat, swirling pan occasionally until perfectly browned* (stir occasionally and be careful not to burn). Set butter aside to cool 25 minutes, or until lukewarm.
Heat oven to 375 degrees. Butter 10 cups in a 12 cup muffin tin and dust lightly with flour, then shake off excess flour.
In a mixing bowl whisk together, flour, baking powder, salt, 1/4 tsp cinnamon and nutmeg for 30 seconds, set aside. In a separate mixing bowl whisk together 1/4 cup + 2 Tbsp granulated sugar and light brown sugar until no clumps of brown sugar remain. Using a spoon, stir in cooled browned butter.
Add in egg and vanilla and mix until well blended. Stir in milk. Make a well in dry mixture and pour wet mixture into well in dry mixture. Fold mixture just until combined (batter will be slightly lumpy).
Divide mixture evenly among 10 prepared cups in muffin tin. Bake in preheated oven until toothpick inserted into center comes out clean, about 15 - 17 minutes. Remove from oven and drop muffins onto a cooling rack. Allow to cool just until cool enough to handle.
While muffins are baking, brown 1/2 cup butter in a small saucepan. Remove from heat and pour into a heat proof bowl and allow to cool slightly. In a separate bowl whisk together 1/2 cup granulated sugar and 1 tsp cinnamon.
Whisk browned butter then dip and roll each muffin into browned butter to evenly coat entire muffin, then transfer to cinnamon sugar mixture and roll to evenly coat. Serve warm. Store in an airtight container at room temperature.