Place butter and oil in a 12-inch cast-iron skillet or a 9-by-13-inch baking pan. Heat in the oven until melted and fragrant, 5 to 7 minutes.
Meanwhile, combine flour, baking powder and salt in a large bowl. Add milk, sugar and vanilla; stir to combine.
Add the melted butter mixture to the batter and stir to combine. Pour the batter into the hot pan. Spoon peaches and blueberries evenly over the batter.
Return the pan to the oven and bake until the top of the cobbler is browned and the batter around the fruit is completely set, 50 minutes to 1 hour. Remove to a wire rack to cool for at least 15 minutes. Serve warm.