For the pierogi filling:
2 tablespoons vegetable oil
1 pound skinless, boneless chicken thighs, chopped into 2-inch pieces
1 large shallot, minced
1 large garlic clove, minced
1 half-inch-thick piece fresh ginger, peeled and minced
1/2 teaspoon crushed red pepper flakes
1 cup chicken broth
1 tablespoon soy sauce
1 tablespoon unseasoned rice vinegar
1 tablespoon light brown sugar
For the scallion dough:
2 large eggs
1/2 cup (4 ounces) sour cream or plain Greek yogurt (full-fat, reduced-fat, or nonfat)
2 tablespoons toasted sesame oil
1 tablespoon unsalted butter, melted and slightly cooled
1 teaspoon kosher salt
1 1/2 cups (6 3/8 ounces) unbleached all-purpose flour
1/2 cup minced scallions, green parts only
1 tablespoon water
For the dipping sauce:
2 tablespoons soy sauce
2 tablespoons unseasoned rice vinegar
2 tablespoons honey
1/2 teaspoon hot chili sesame oil or garlic-chili paste
1 minced scallion, green parts only (optional)
Make the filling: Heat the oil in a medium (3- to 4-quart) Dutch oven or heavy pot over medium-high heat. Add the chicken and cook until browned, about 5 minutes. Flip the chicken and stir in the shallot, garlic, ginger, and pepper flakes. Cook about 5 minutes more.
Stir in the broth, soy sauce, vinegar, and sugar, and bring the liquid to a simmer. Reduce the heat to medium-low, cover, and simmer for about 45 minutes, until the chicken is very tender.
Uncover and take a look: If the liquid hasn’t fully reduced to a thick and sticky sauce, continue simmering for about 5 to 10 minutes more.
Remove from the heat and let the chicken cool slightly, then shred the chicken and sauce with 2 forks, or place into a stand mixer bowl and shred with the paddle attachment on low speed. (The filling can be made up to 2 days ahead. Cover and refrigerate until ready to assemble.)
Make the dough: Whisk 1 of the eggs, sour cream or yogurt, sesame oil, butter, and salt in a bowl. Pour the flour into a large bowl. Gently stir wet ingredients into flour. The dough will initially be very dry and shaggy, seeming as if it will never come together, but have no fear: Keep stirring, and it will pull itself into shape.
Once the dough starts to come together, add the scallions. Press and smash the dough against the sides of the bowl with your palms, picking up dough bits and essentially kneading it within the bowl until it forms a ball.
Tip dough and any remaining shaggy flakes