Preheat the oven to 350 degrees F (175 degrees C). Spray doughnut pan with non-stick cooking spray.
Combine eggs, pumpkin, brown sugar, sugar, butter, oil, and pumpkin pie spice extract in a bowl; mix until well combined.
Sift flour, pumpkin pie spice, baking powder, baking soda, and salt together in a second bowl. Add to egg mixture and stir until combined. Evenly distribute mixture amongst the 12 donut pans. Wet finger and smooth out the tops.
Bake in the preheated oven until a toothpick inserted in the center comes out clean, 12 to 16 minutes. Cool completely on wire racks, at least 30 minutes.
Stir together powdered sugar, 1/4 cup milk, vanilla, pumpkin pie spice extract, and salt in a bowl until completely combined. Add more milk, 1 tablespoon at a time, if necessary to reach desired glaze consistency.
Dip one side of the doughnuts into the glaze and place on a rack over a cookie sheet to allow glaze to set.