Preheat oven to 400°.
To prepare filling, pierce the potatoes with a fork; place potatoes on a baking sheet. Remove white papery skin from garlic head (do not peel or separate cloves). Drizzle garlic head with oil, and wrap in foil. Place garlic on baking sheet. Bake potato and garlic at 400° for 1 hour or until tender. Cool slightly. Peel potatoes; discard skins. Place potatoes in a large bowl. Separate garlic cloves; squeeze to extract garlic pulp. Add garlic to potatoes, and mash with a potato masher until smooth. Stir in sour cream, 1/2 teaspoon salt, and black pepper. Set aside.
To prepare dough, combine the yogurt, 1/2 teaspoon salt, and eggs in a large bowl. Lightly spoon flour into dry measuring cups; level with a knife. Add flour to yogurt mixture; stir well to combine. Turn dough out onto a lightly floured surface; knead until smooth. Divide dough into 24 equal portions, shaping each into a ball. Working with 1 portion at a time (cover remaining dough to prevent drying), roll each ball into a 3 1/2-inch circle on a lightly floured surface. Spoon 1 rounded tablespoon filling onto half of each circle. Fold dough over filling; press edges together to seal. Repeat procedure with remaining dough and filling.
Bring 12 cups water to a boil in a large Dutch oven. Add half of pierogi; cook 6 minutes or until tender. Remove pierogi with a slotted spoon; place in a colander. Repeat procedure with the remaining pierogi.
To prepare the browned butter while pierogi cook, melt butter in a large nonstick skillet over medium heat; cook 10 minutes or until lightly browned (be careful not to burn). Add shallots; cook 1 minute. Add pierogi; cook 2 minutes, tossing to coat.