Start the sauce first. Heat the 2 tablespoons oil in a large pot over medium heat. Add the onions with a pinch of salt and cook, stirring often, for 15 to 20 minutes, until soft, sweet, and light golden.
Add the carrots, potatoes, garlic, and ginger. Cook 2 minutes, stirring. Pour in the chicken stock, bring to a simmer, and cook 12 to 15 minutes, until the potatoes are just tender.
Make the roux in a small pan: melt the butter over medium heat, stir in the flour and curry powder, and cook, stirring constantly, for 2 minutes, until warm, toasty, and a shade darker. Take off the heat.
Whisk about 1 cup of hot stock from the pot into the roux until smooth, then pour it back into the pot. Simmer 5 minutes, stirring often, until thick and glossy. Stir in the honey or apple, soy sauce, and Worcestershire. Taste for salt and keep warm over low heat.
Pound chicken breasts between sheets of plastic to an even 1/2-inch thickness. Season both sides with salt and pepper.
Set up flour, beaten eggs, and panko in three shallow bowls. Coat each piece in flour, then egg, then press firmly into panko on both sides. Rest coated cutlets 5 minutes.
Heat about 1/3 inch of oil in a large skillet over medium-high heat to about 350°F (175°C). Fry cutlets in batches, 3 to 4 minutes per side, until deep golden and cooked through (165°F / 74°C). Drain on a rack and rest 3 minutes, then slice into 3/4-inch strips.
Spoon rice onto one side of each plate, ladle curry over the rice, then lay the sliced cutlet across the border so half stays out of the sauce. Add pickles and scallions and serve right away.