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Step 1
Heat 1 Tbsp vegetable oil in a large non-stick skillet over medium-high heat. Add carrots and saute 3 minutes then add green onions, garlic and ginger and saute 2 minutes longer, set aside.
Step 2
Cover chicken with plastic wrap and pound to an even thickness using the flat side of a meat mallet. Heat 1 Tbsp olive oil in a large pot over medium-high heat.
Step 3
Season both sides of chicken lightly with salt and pepper, then add to pot and cook until browned on both sides, about 2 1/2 minutes per side.
Step 4
Pour in chicken broth, soy sauce, mirin, rice vinegar, Sriracha, and sesame oil, then pour carrot mixture into soup mixture.
Step 5
Bring to a boil then reduce heat to medium-low, cover and simmer until chicken has cooked through (center should register 165 degrees on an instant read thermometer), about 5 - 7 minutes.
Step 6
Remove chicken breasts from soup and transfer to a cutting board and let rest 5 minutes then cut into strips.
Step 7
Meanwhile, stir in sugar, cabbage and mushrooms and return soup to a boil. Add noodles and cook 3 - 5 minutes longer until noodles are tender. Stir in chicken. Serve warm sprinkled with cilantro.