Combine 1 cup water; 1 cup soy sauce; ⅓ cup dark soy sauce; 2 tablespoons Shaoxing wine; 2 tablespoons packed brown sugar; 1 star anise pod; 1 cinnamon stick; and 1 piece ginger, peeled and halved crosswise, in 10-inch skillet, stirring to dissolve sugar. Place 6 trimmed bone-in chicken thighs skin side up in poaching liquid.

Bring to simmer over medium-high heat. Cover, reduce heat to medium-low, and simmer for 10 minutes. Flip chicken skin side down, reduce heat to low, and continue to simmer, covered, until thighs register 175 degrees, 10 to 15 minutes longer. Transfer chicken to cutting board. Measure out and reserve 1 cup chicken poaching liquid; set aside.

Meanwhile, bring 2 quarts water to boil in large saucepan. Add 1 pound fresh Chinese wheat noodles to boiling water; reduce heat to maintain very gentle simmer; and cook, stirring occasionally, until almost tender (center of noodles should be slightly firm), 3 to 5 minutes. Drain noodles well in colander and rinse with running water.

Bring 6 cups chicken broth to boil in now-empty saucepan over medium-high heat. Add 4 heads baby bok choy, quartered, and cook until tender, about 2 minutes. Transfer bok choy to plate, then whisk ⅔ cup reserved poaching liquid into broth in saucepan. Carve chicken from bone and slice ½ inch thick; discard bones.

Divide noodles among 4 individual serving bowls. Pour broth over top, then add chicken and bok choy. Cut 2 trimmed scallions crosswise into 1-inch pieces. Then slice thin lengthwise. Sprinkle soup with sliced scallions and drizzle with sesame oil. Serve, seasoning with remaining reserved poaching liquid to taste.