Pat the chicken breasts dry with paper towels. Season both sides evenly with kosher salt and black pepper.
Heat the olive oil in a large skillet over medium-high heat until shimmering.
Add the chicken breasts to the skillet and cook for 5 to 6 minutes on one side without moving, until golden brown.
Flip the chicken breasts and add the minced garlic and chopped rosemary to the skillet around the chicken.
Cook for another 5 minutes, stirring the garlic and rosemary occasionally to prevent burning.
Add the lemon zest, lemon juice, and chicken broth to the skillet. Bring the liquid to a simmer.
Reduce the heat to medium-low and cook for an additional 5 to 7 minutes, spooning the sauce over the chicken occasionally, until the chicken is cooked through and reaches an internal temperature of 165°F (74°C).
Remove the chicken breasts to a plate and tent with foil to keep warm.
Stir the unsalted butter into the skillet sauce until melted and smooth.
Pour the sauce over the chicken breasts and serve immediately.