Place ramen bricks in a shallow heat-safe bowl. Pour boiling water over ramen and let stand until noodles start to separate and soften.
Mix soy sauce, balsamic vinegar, sesame oil, sugar, garlic, and ginger together in a small bowl or glass jar.
Heat vegetable oil in a large skillet or wok over medium-high heat. Saute bok choy, onion, and carrots until onion is translucent, 3 to 5 minutes. Add cremini mushrooms, shiitake mushrooms, and kale. Saute until kale begins to wilt, 3 to 5 minutes.
Separate ramen noodles with a fork and drain any excess water. Add drained noodles to the skillet and increase heat to high. Pour soy and vinegar sauce over the noodles and vegetables. Stir until flavors combine, about 2 minutes.