Pound chicken breasts to an even 1/2-inch thickness. Season both sides with salt and pepper.
Set flour, beaten eggs, and panko in three shallow dishes. Coat each cutlet in flour, egg, then panko; press crumbs gently so they adhere.
For the curry, heat 1 tablespoon oil in a saucepan. Cook onion and carrot for 5 minutes, add broth, and simmer 12–15 minutes until tender.
Lower heat and stir in curry roux cubes and soy sauce until dissolved and thick. Keep warm on low.
Heat frying oil in a deep skillet to about 350°F (175°C). Fry cutlets in batches for 3–4 minutes per side until golden and 165°F (74°C) inside. Drain on a rack.
Slice chicken and serve with rice and curry sauce alongside or over the rice.