Pat the chicken dry. If using chicken breasts, halve them crosswise to create two smaller pieces.
Season both sides of the chicken with salt and pepper.
Heat the oil in a large, high-sided pan (with a tight-fitting lid) over medium-high heat.
Working in batches to prevent crowding the pan, add the chicken and sear until golden brown on both sides.
Transfer the chicken to a plate.
Melt the butter in the same pan over medium-high heat.
Add the onion and carrots and cook for 3 to 5 minutes, until soft.
Add the garlic, Italian seasoning, and paprika and cook for 30 seconds, until fragrant.
Add the rice and cook for 30 seconds, until lightly toasted, stirring constantly.
Add the broth and bring to a simmer, scraping up any browned bits from the bottom of the pan.
Return the chicken to the pan with any accumulated juices.
Bring the broth to a simmer.
Reduce the heat to low.
Cover and cook for 15 to 20 minutes, until the rice is tender and the liquid is absorbed. If liquid is absorbed before 15 minutes, add more as needed, no more than ¼ cup at a time.
Remove the pan from the heat and fold in the Pecorino Romano (or parmesan) cheese.
Season to taste with salt and pepper.
Top with lemon slices, basil and/or green onions (if using) and serve.