Preheat oven to 400 degrees. Line 15 or 16 muffin cups with paper liners or spray with cooking spray. Set pan aside.

Combine the eggs, sugar, pumpkin puree, vegetable oil, and vanilla in a large mixing bowl. Beat until smooth.

Combine the flour, baking soda, baking powder, salt, and spices in a medium mixing bowl and whisk to combine. Gradually mix the dry ingredients into the pumpkin-oil mixture until just moistened.

Fill prepared muffin cups about 3/4 full. Bake 15 to 18 minutes, until a toothpick inserted into the center of the muffins comes out clean. Cool in pan for 5 minutes, then remove to a wire rack to cool completely before glazing.

Combine the powdered sugar, espresso powder, melted butter, and 3 teaspoons of the milk in a small mixing bowl. Stir vigorously until smooth. Add additional coffee or milk if necessary to thin the glaze. Spoon over the muffins, then let the muffins sit until the glaze sets. Serve.