In a large saucepan with a tight-fitting lid over medium heat, melt the butter.
Add the onion and cook until softened, about 3 to 4 minutes.
Stir in the garlic and cook for 30 seconds.
Add the uncooked rice and stir to coat. Toast for 1 minute.
Pour in the chicken broth and bring to a boil. Once boiling, reduce heat to low, cover, and simmer for 18 minutes.
Remove from heat and let sit, covered, for 5 minutes. Fluff rice gently with a fork, then keep covered.
While the rice cooks, pat the chicken dry and season evenly with kosher salt, black pepper, and paprika.
In a large skillet over medium-high heat, heat the olive oil. Add the chicken in a single layer and cook for 5 to 6 minutes, flipping halfway through, until golden brown and cooked through to an internal temperature of 165°F. (Depending on the size of your skillet, you may have to work in batches.) Transfer cooked chicken to a plate and tent loosely with foil.
Reduce the heat to medium-low. Add the heavy whipping cream and Dijon mustard to the skillet and stir to combine. Bring to a gentle simmer and cook for 2 to 3 minutes, until slightly thickened.
Add the cowboy butter, stirring until fully melted and incorporated.
Stir in the Parmesan cheese and cook just until smooth.
Add the cooked rice to the skillet and fold gently until evenly coated and creamy.
Fold in the chicken. If needed, add a splash of warm broth or cream to loosen slightly.
Garnish with chopped parsley and serve immediately. Finish with a squeeze of fresh lemon, if desired.