Slice the chicken breasts into thin 1/2-inch strips and pat them dry with paper towels—this helps them brown better and cook more evenly. Mince the garlic, grate the ginger, thinly slice the onion into half-moons, and shred the cabbage into 1/4-inch ribbons. Season the chicken strips generously with salt and pepper on both sides. I like to season the chicken at least 5 minutes before cooking so the salt can penetrate the meat and improve browning.
Heat the olive oil in a large skillet over medium-high heat until it shimmers. Add the seasoned chicken strips in a single layer and cook for 5-7 minutes, stirring occasionally, until they're golden brown and cooked through. The chicken won't be fully cooked yet, and that's okay—it will finish cooking when we add it back to the pan later. Transfer the chicken to a clean plate and set aside.
Reduce heat to medium and add the butter to the same skillet. Once melted and foaming, add the sliced onions and cook for 3 minutes, stirring occasionally, until they begin to soften and turn translucent. Add the minced garlic and grated ginger, stirring constantly for about 30 seconds until fragrant. This technique of blooming the aromatics in fat releases their essential oils and creates a deep flavor foundation for the dish.
Add the shredded cabbage and paprika to the pan, stirring to coat everything evenly with the buttery aromatics. Cook for 10-15 minutes over medium heat, stirring occasionally, until the cabbage is tender and lightly caramelized at the edges. The cabbage will release its moisture and soften down significantly. I find that letting the cabbage sit undisturbed for 2-3 minutes between stirs helps it develop a slight golden color and deeper flavor.
Return the seared chicken to the skillet along with the soy sauce, stirring gently to combine and heat the chicken through, about 2-3 minutes. Taste the dish and adjust seasoning with additional salt and pepper as needed—the soy sauce adds saltiness, so taste before adding more. Serve immediately while everything is hot and the cabbage still has a bit of textural integrity.