In a large bowl, beat eggs, sugar and butter until smooth. Beat in pumpkin and lemon juice.
Combine flour, baking powder, baking soda, salt, cinnamon and nutmeg; gradually add to pumpkin mixture alternately with milk. Cover and refrigerate for 2 hours.
Turn onto a lightly floured surface; knead 5-6 times.
Roll to 3/8-in. thickness.
Cut with a lightly floured 2-1/2-in. doughnut cutter.
In an electric skillet or deep-fat fryer, heat oil to 375°. Fry , a few at a time, until golden brown on both sides.
Drain on paper towels.
Combine frosting ingredients; spread over cooled doughnuts.