Preheat oven to 350 degrees F.
In a large bowl, combine 1 cup granulated sugar and 1 cup vegetable or canola oil until blended.
Add three eggs, one at a time, beating well after each addition.
In a separate bowl, combine 3 cups all-purpose flour, 2 teaspoons baking soda, 2 1/2 teaspoons pumpkin pie spice, 1/2 teaspoon salt, and one package instant vanilla pudding mix.
Add flour mix to egg mixture alternately with one can (15 ounces) pumpkin puree, beating well after each addition.
Fill lined cupcake pan three-quarters full, and bake at 350 degrees F for 25 to 30 minutes or until toothpick inserted near the center comes out clean. Remove from pan, and cool completely.
For me, 25 minutes was the perfect amount of time. I also recommend using an ice cream scoop to fill your cupcake liners. It makes it easy to fill each cupcake liner with the same amount of mixture. I have also seen a huge improvement in the look of my cupcakes since I have been using my ice cream scoop.
Whisk together 2 cups powdered sugar and 1/2 teaspoon cinnamon, and add 1 tablespoon of milk at a time until desired consistency. I used about 6 to 7 tablespoons of milk.
Dip warm cupcakes into glaze to coat, then enjoy some warm pumpkin cupcakes.