Place the rice, water, 1 teaspoon butter, and a pinch of salt in a medium saucepan and bring to a boil over high heat. Reduce the heat, cover the pot, and continue to simmer for 20 minutes. Remove the pot from the heat, leaving it covered, and set it sit for 5 minutes. Remove the lid and fluff the rice before transferring to a serving bowl.

While the rice is cooking, season the chicken on both sides with salt and pepper. Heat a a very large skillet over medium-high heat and add 2 teaspoons of the olive oil. When the oil is hot, add three of the chicken pieces and cook until golden brown and almost cooked through, about 3 minutes per side. Transfer the chicken to a plate, and repeat with the remaining oil and the remaining 3 cutlets. Tent the plate of cutlets with foil to keep them warm while you make the sauce.

Return the skillet, without cleaning it, to medium heat. Add the leeks and garlic, and sauté for 4 minutes, until the leeks start to become tender. Stir in the tomatoes and thyme and cook for another minute until the tomatoes start to soften. Add the white wine, (the alcohol will burn off in the cooking process, so it’s fine for kids), and stir to scrape up any browned bits from the bottom. Add the chicken broth and bring to a simmer, and cook for about 4 minutes, until the liquid reduces by about half.

Add the slightly undercooked chicken to the pan and let the pieces simmer to finish cooking, 2 to 3 minutes. Place the chicken on a serving plate, or individual plates, and add the lemon juice and the remaining tablespoon of the butter to the pan and stir until the butter is melted. Season with salt and pepper to taste.

Pour the sauce over the chicken, and serve the sauced chicken with the hot rice.