Combine flour, water, and ½ teaspoon salt in a bowl; mix with a wooden spoon until dough comes together in a sticky ball. Sprinkle in more flour as needed and knead by hand until soft and supple. Wrap in plastic; rest at room temperature for 1 hour.

Combine ground pork, green onions, chives, garlic, and ginger in a bowl. Season with 2 tablespoons soy sauce, 4 teaspoons salt, 2 teaspoons gochugaru, sesame oil, and black pepper; mix filling using two forks until just combined. Cover the bowl with plastic; refrigerate until ready to use.

Whisk vinegar, 3 tablespoons soy sauce, and 1 teaspoon gochugaru, if using, into chicken broth; set aside.

Divide dough into 4 equal pieces. Roll 1 piece into a rope about the thickness of your thumb; flatten into a long, thin band about 3 inches wide using a rolling pin, flouring only as needed. Dust top with flour; flip over. Dip your fingers in some water and lightly brush the surface until slightly sticky.

Arrange about ¼ pork filling down middle of dough about the thickness of your thumb using moistened fingers. Fold entire length of dough over filling, pressing down to seal dough. Trim excess dough if desired. Dust top with a little more flour; flip over so seam side is on bottom. Divide into 16 pieces using the edge of your hand; cut with a bench scraper. Repeat with remaining dough and filling.

Heat oil in a skillet over high heat. Cook dumplings, in batches, seam-sides up, until golden brown, 1 to 2 minutes per side. Add a portion of broth; bring to a boil. Cover and continue to boil until pork is no longer pink, about 3 minutes. Serve dumplings in broth in bowls.