Preheat the oven to 190°C / 375°F.
In a saucepan, bring water to a boil. Stir in rice and salt and return to a boil over medium-high heat.
Reduce heat to a simmer, cover, and cook until rice is tender and has absorbed all the liquid, about 16 to 18 minutes.
Remove from heat and let steam, covered, for 10 minutes.
Working one at a time, place chicken between sheets of parchment paper and pound with a mallet or rolling pin until about .635 cm / ¼" thick.
Season chicken with salt and black pepper.
Arrange chicken smooth side down. Cover one half of each breast with one slice of ham and one slice of cheese, leaving space around the edges. Fold breast in half over ham and cheese and press edges together to seal.
In a bowl, combine bread crumbs, parsley, salt, and black pepper. Whisk egg and water in another bowl. Spread flour on a plate.
Press both sides of chicken in flour, shaking off excess, then dip both sides into egg mixture. Coat with bread crumb mixture, patting with fingers to adhere. Transfer to a plate.
In a large skillet, heat oil over medium-high until shimmering. Cook chicken, turning occasionally, until browned on all sides, 2 to 3 minutes per side. Transfer to a wire rack set in a baking sheet; season with salt. Bake chicken until an instant-read thermometer inserted into the thickest part registers 75°C / 165°F, about 5 minutes.
Using a fork, fluff the rice. Serve with chicken.