Preheat oven to 350°F (175°C). Place butter in the baking pan and let it melt as the oven heats for a buttery, non-stick base.
Cut each biscuit into quarters. Mix both sugars with cinnamon, then toss biscuit pieces until coated.
Remove the pan, spread the coated biscuits evenly, and bake for 15 minutes until golden and puffed.
Whisk powdered sugar, milk, and vanilla until smooth, adjusting thickness as desired.
Cool biscuits for a minute, drizzle icing over while warm, and serve soft, sticky, and delicious straight from the pan.