Heat oven to 375 degrees F. Line a 12-cup muffin pan with paper baking cups.

In a large bowl or bowl of a stand mixer, beat sugars, oil, butter and vanilla until combined and smooth. Beat in eggs one at a time. Add baking powder and baking soda; stir to combine.

Combine coffee and milk in a measuring cup. Alternately add flour and milk-coffee mixture to bowl, beginning and ending with the flour, stirring after each addition until combined.

Divide batter among paper baking cups. Bake 20 to 25 minutes until a toothpick inserted in the center comes out clean. Transfer muffins to a cooling rack to cool completely.

Meanwhile, make the glaze: In a medium bowl, stir together powdered sugar, vanilla and enough milk to form a thick but drizzly glaze. Dip fully cooled muffins in the glaze. Let excess glaze drip off for a few seconds, then place muffins back on cooling rack so the glaze drips down the sides as it cools and sets.

Once the glaze is set, drizzle caramel sauce over the tops of the muffins.