Combine the olive oil and 1 tbsp of the butter in a large pot (I use an 8 quart pot) over medium heat. While the pot heats, pat the chicken cutlets dry all over and season with salt and pepper. Add the chicken to the pot and brown, about 3 minutes per side (the chicken doesn’t need to cook all the way through now because it will later!) Transfer the browned chicken to a cutting board, leaving any drippings in the pot. When cool enough to handle, cut the chicken into half-inch pieces and refrigerate.
While the chicken cools, add the remaining 1 tbsp butter to the pot along with the onion, carrot, celery, garlic, 1/2 tsp salt, and black pepper to taste. Sweat until the onions are just translucent, about 8-10 minutes.
Add the rice to the pot and stir to combine. Cook about 3 minutes to toast the rice, stirring occasionally. Then, sprinkle the flour and thyme over the top and stir to combine. Cook, stirring, about 1 minute. Stir in the white wine and simmer about 3 minutes, until mostly absorbed.
Add the chicken stock and scrape any bits off the bottom of the pot to incorporate. Bring to a boil over high heat, then reduce heat to low and cover. Cook 25 minutes, or until rice is cooked through. Taste for salt and adjust if needed.
Stir in the browned chicken and simmer 5-8 minutes more, or until chicken is cooked and warmed through.
Stir in the lemon juice and parsley just before serving. Enjoy!