Preheat the oven to 350 degrees F (175 degrees C). Butter the bottom of an 8-inch round or square baking pan and sprinkle with 2 tablespoons brown sugar.

Combine white sugar, cinnamon, 2 tablespoons brown sugar, and vanilla extract in a bowl. Mix filling well.

Combine confectioners' sugar and milk for glaze in another bowl until well mixed.

Separate biscuits and roll or punch down each one until flat. Coat with melted butter and sprinkle each with filling mixture to coat. Roll up each piece of dough and cut in 1/2, placing each cut-side down into the prepared pan, yielding 20 pieces.

Bake in the preheated oven until biscuit portion on top of rolls begins to brown, about 15 minutes. Remove from the oven and glaze.