Place potato in a medium saucepan; cover with cold water. Bring to a boil. Reduce heat, and simmer 20 minutes; drain. Combine potato, cheddar cheese, 2 teaspoons butter, and 1/4 teaspoon salt in a bowl; mash with a potato masher.
Weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour and remaining 1/2 teaspoon salt in a medium bowl. Combine 1/2 cup sour cream and eggs in a small bowl, stirring with a whisk. Add sour cream mixture to flour mixture, stirring just until combined. Wrap dough in plastic wrap; refrigerate 30 minutes.
Heat a small skillet over medium-high heat. Add oil to pan; swirl to coat. Add onion and garlic; sauté 4 minutes. Stir onion mixture into potato mixture.
Divide dough into 2 portions. Roll each portion to a 1/8-inch thickness on a lightly floured surface; cut with a 4-inch round cutter into 18 rounds. Working with 1 round at a time, spoon 1 tablespoon potato mixture onto each round. Fold dough over filling; press edges together to seal. Repeat with remaining dough and potato mixture.
Bring 12 cups water to a boil in a large saucepan. Add half of pierogies; cook 2 minutes or until pierogies float. Remove cooked pierogies with a slotted spoon. Repeat procedure with remaining pierogies.
Melt 2 teaspoons butter in a large nonstick skillet over medium-high heat. Add half of pierogies to pan; cook 2 minutes on each side or until golden brown. Remove from pan. Repeat procedure with remaining 2 teaspoons butter and remaining pierogies. Place 3 pierogies on each of 6 plates; top each serving with 2 teaspoons sour cream. Sprinkle evenly with parsley.
MyRecipes is working with Let's Move!, the Partnership for a Healthier America, and USDA's MyPlate to give anyone looking for healthier options access to a trove of recipes that will help them create healthy, tasty plates. For more information about creating a healthy plate, visit www.choosemyplate.gov.