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1 c. whole milk
1/4 c. plus 1 tsp. granulated sugar, divided
1 packet (or 2 1/4 tsp.) active dry yeast
4 1/2 c. all-purpose flour, plus more for surface
1/2 tsp. kosher salt
6 tbsp. melted butter
2 large eggs
1/2 tsp. pure vanilla extract
Canola or vegetable oil, for frying
1/4 c. whole milk
2 c. powdered sugar
1/2 tsp. pure vanilla extract