Preheat the oven to 350 degrees F and line a muffin tin with 4 paper liners.
In a medium bowl, whisk together the flour, pumpkin pie spice, baking powder, baking soda, and salt.
In a separate bowl, mix the granulated sugar, pumpkin puree, egg, and vegetable oil until smooth.
Pour the wet mixture into the dry ingredients and stir until just combined. Do not overmix.
Divide the batter evenly among the 4 liners and bake for 18 to 20 minutes, or until a toothpick comes out clean. Let the cupcakes cool completely.
While the cupcakes cool, melt the butter in a small saucepan over medium heat. Cook it until it turns golden brown and smells nutty, then remove from heat and let it cool slightly.
In a mixing bowl, beat the browned butter with the powdered sugar, vanilla extract, and milk until smooth and creamy.
Spread or pipe the frosting onto the cooled cupcakes.