Heat 1 Tbsp vegetable oil in a large non-stick skillet over medium-high heat. Add diced carrots and sauté for about 3 minutes until they start to soften, then stir in sliced green onions, minced garlic, and ginger, continuing to sauté for an additional 2 minutes until fragrant.
Cover the chicken breasts with plastic wrap and pound them to an even thickness using a meat mallet's flat side; this ensures even cooking throughout. In a large pot over medium-high heat, add another tablespoon of olive oil.
Season both sides of the chicken breasts lightly with salt and freshly ground pepper before adding them to the hot pot. Cook until they are browned on both sides, approximately 2 ½ minutes per side, before proceeding with the rest of the recipe.
Pour in the low-sodium chicken broth along with soy sauce, mirin, rice vinegar, Sriracha, and sesame oil into the pot with chicken; then add your sautéed carrot mixture into this delicious blend.
Bring everything to a boil before reducing heat to medium-low; cover the pot and simmer until the chicken is fully cooked through (the center should register at least 165 degrees). This should take about 5 -7 minutes depending on thickness.
Remove cooked chicken from the pot and let it rest on a cutting board for about 5 minutes before slicing it into strips. Stir in granulated sugar, chopped Napa cabbage, sliced crimini mushrooms back into your boiling soup, then add dry ramen noodles and cook for an additional 3-5 minutes until noodles are tender; return sliced chicken back into soup just before serving warm garnished with fresh cilantro sprinkled on top.