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.
Cut chicken breasts in half lengthwise to make four thinner cutlets. Coat them in flour.
In a large skillet, heat olive oil and butter over medium-high heat. Once pan is hot, add the chicken. Cook it until golden on the outside, about 2 minutes per side. Remove chicken from pan and set aside.
In the same pan, melt butter. Add mushrooms and oregano. Once mushrooms start to release water, about 3 minutes, add garlic, Continue cooking the mushrooms until all the water is cooked off, about 2 more minutes. Remove mushrooms from pan and set aside.
In the same pan, add chicken broth, lemon juice, and Dijon mustard. Stir until mustard dissolves, and let it reduce for 3 to 4 minutes.
Add heavy cream, chicken and mushrooms to the pan. Let chicken cook until it reaches an internal temperature of 75°C / 165°F, and the sauce has thickened. Season with salt and black pepper.
Using a fork, fluff the rice. Serve with chicken.