Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes.
Drain.
Heat a large skillet over medium-high heat. Cook and stir beef in the hot skillet until browned and crumbly, 5 to 7 minutes; drain in a sieve or colander and rinse off excess fat with hot water.
Mash potatoes with milk in a bowl until smooth; stir in onion, garlic, 1 pinch salt, and pepper.
Mix ground beef into potato mixture until filling is evenly combined.
Whisk flour and 4 teaspoons salt together in a deep bowl; stir in 1 cup water and eggs until dough is soft. Turn dough onto a floured surface and knead until smooth, 1 to 2 minutes. Divide dough into 4 equal parts and arrange on a baking sheet; cover with a damp towel or plastic wrap and let stand for 10 minutes.
Roll each piece of dough on a floured work surface until thin; cut into rounds using the rim of a large-mouth cup or large cookie cutter.
Spoon 1 to 2 tablespoons filling onto each round. Fold dough over filling creating a 1/2 circle; press edges together, sealing with a dash of water if needed. Arrange varenyky in a single layer on plates and cover with a clean towel.
Bring a pot of lightly salted water to a boil; cook 2 to 3 varenyky at a time in boiling water until cooked through and dough is puffed, about 4 minutes.
Remove varenyky with a slotted spoon and repeat with remaining varenyky.
