Heat 1 tablespoon oil in a large Dutch oven over medium heat. Add chicken pieces; cook, stirring often, until no longer pink, about 5 minutes. Add 1 tablespoon garlic and 2 teaspoons ginger; cook, stirring constantly, until fragrant, about 1 minute.

Add 1 tablespoon miso and ½ teaspoon salt; stir to coat. Stir in 8 cups broth; bring to a boil over high heat. Add 3 cups noodles and 1 cup frozen vegetables; return to a simmer over medium-high heat. Reduce heat to medium to maintain simmer; cook, stirring occasionally, until the noodles are tender and the chicken is cooked through, about 8 minutes. Stir in 2 tablespoons scallions and ¼ teaspoon crushed red pepper.