Heat 2 tbsp. oil in a large frying pan over medium-high heat. Season chicken with 1/2 tsp. salt and 1/4 tsp. pepper and arrange in pan. Brown, turning once, 3 to 4 minutes per side. Transfer chicken to a plate. Drain fat and return pan to stovetop.
Add rice to pan and cook, stirring, until glossy, 2 minutes. Add broth, stirring to combine. Cover and bring to a boil, then reduce heat to a simmer.
Set chicken on top of rice, cover, and cook until rice is tender and has absorbed most of the liquid, about 15 minutes.
Meanwhile, whirl together remaining 1/4 cup oil, the lime juice, parsley, basil, cilantro, green onions, garlic, chile flakes, 2 tbsp. water, and remaining 1/2 tsp. salt and 1/4 tsp. pepper in a blender until the consistency of salsa, about 1 minute.
Transfer chicken to a plate. Stir half the chimichurri into rice. Serve rice with chicken and more chimichurri.