Combine rinsed rice, water or broth, and salt in a medium saucepan. Bring to a boil, cover, reduce heat to low, and cook for 15 minutes. Remove from heat and rest, covered, for 10 minutes before fluffing.

Pat chicken cutlets dry. Mix paprika, garlic powder, salt, and black pepper, then season both sides evenly.

Heat olive oil in a large skillet over medium-high. Sear chicken for 3 to 4 minutes per side until golden. Work in batches if needed. Transfer to a plate; the chicken will finish cooking in the sauce.

Reduce heat to medium and melt butter in the skillet. Add onion and mushrooms. Cook for 6 to 8 minutes, stirring occasionally, until tender and browned. Add garlic and thyme and cook for 30 seconds.

Sprinkle flour over the vegetables and stir for 1 minute. Gradually whisk in chicken broth, scraping the browned bits from the pan. Stir in half-and-half and Dijon. Simmer for 3 to 4 minutes until the gravy lightly thickens.

Return chicken and its juices to the skillet. Spoon gravy over the cutlets, cover, and simmer gently for 5 to 8 minutes, until the chicken reaches 165°F in the thickest part.

Taste the gravy and adjust seasoning. Divide rice among plates, top with chicken, and spoon over the mushroom-onion gravy. Garnish with parsley and black pepper.