Prepare the chicken: Season the chicken with salt and pepper on both sides. If using chicken breast, cut it into two thinner filets and pound to even thinness for faster, crispier frying.
Set up the breading station: In one bowl, whisk together the flour, salt, pepper, and garlic powder. In a second bowl, whisk together the eggs, milk, salt, and pepper. In a third, mix the panko breadcrumbs with salt, pepper, and garlic powder.
Make the broth: In a small bowl, whisk together chicken broth, soy sauce, rice vinegar, cornstarch, and brown sugar until smooth. Set aside.
Caramelize the onions: Heat a pan over medium-low heat and drizzle in a little neutral oil. Add the sliced onions and cook slowly until golden and soft, stirring occasionally. Add splashes of water as needed to deglaze and help them caramelize evenly. Once sweet and jammy, turn off the heat and set aside.
Preheat the oil: In a large skillet, heat about an inch of neutral oil over medium heat until hot and shimmering.
Bread the chicken: Dredge each piece first in the flour, then in the egg mixture, and finally coat generously with panko breadcrumbs, pressing lightly so they stick well. To make the process less messy, keep one hand for the dry ingredients and the other for the wet.
Fry the chicken: Fry until golden and crispy on both sides, about 3 to 4 minutes per side depending on thickness. Transfer to a wire rack to rest while you prepare the final step.
Simmer the onions and sauce: In a separate pan — or in the same one you used to caramelize the onions — add a single portion of the caramelized onions along with some of the broth mixture. Let it come to a gentle simmer over medium heat. You can also cook two portions side by side in a large pan to be more efficient.
Add the chicken and egg: Slice one crispy chicken cutlet into strips and place it in the center of the pan over the onions and broth. Whisk one egg in a small bowl and pour it evenly over the top. Let it cook over medium heat until the egg bubbles and sets softly around the chicken, about 1 to 2 minutes. Repeat this step for the second portion.
Assemble and serve: Scoop warm jasmine rice into bowls, then carefully slide the chicken, egg, and onions on top. Spoon any leftover broth from the pan over the rice and finish with sliced green onions. Serve immediately.