1. In a 5- to 6-quart Dutch oven, cook the chicken, skin side down over medium heat, turning once, until the fat renders, about 10 minutes. Carefully pour 2 tablespoons rendered fat into a bowl. Discard the remaining fat and the chicken skin. Reserve the chicken.
2. In a large bowl, beat the eggs with an electric mixer on medium speed until pale and foamy. Reduce the speed to low and beat in the reserved chicken fat, chives, and salt. With a spatula, stir in the matzo meal just until incorporated, then stir in the seltzer. Cover and refrigerate while you prepare the soup ingredients, at least 15 minutes.
3. Trim the radishes and cut the larger ones in quarters and smaller ones in halves. Bring a large pot of salted water to a boil. Add the radishes and simmer for 25 minutes.
4. Meanwhile, return the chicken thighs in the pot to high heat and add the broth and 2 cups water. Bring to a boil, then reduce the heat and simmer 10 minutes. Meanwhile, form the matzo mixture into 1/2-inch balls with damp hands. Carefully drop them into the broth with the chicken and simmer until cooked through, about 15 minutes. With a slotted spoon or tongs, transfer the chicken to a plate, cool slightly, then pull the meat in large shreds and return to the soup. Discard the bones.
5. With the same slotted spoon transfer the radishes to the soup. Garnish with chives.