Slice and core the jalapeño, then cut into small pieces. Place in a bowl with the cream cheese and grated cheddar cheese. Mix until combined.
Beat 1 egg in a shallow dish. Set up two more shallow dishes: one with flour, one with breadcrumbs.
Slice each chicken breast horizontally to create a pocket, cutting about halfway through. Fill each pocket with roughly 1-2 tablespoons of the jalapeño cheese mixture.
Coat each filled chicken breast in flour, shaking off excess. Dip in the beaten egg. Coat thoroughly with breadcrumbs.
Heat a thin layer of oil in a large pan over medium-high heat. Once hot, fry the breaded chicken for 3-4 minutes per side until golden brown.
While frying, preheat the oven to 180°C / 356°F (fan oven setting). Once the chicken is golden, transfer it to the hot oven and bake for 20 minutes.