Pat the chicken breasts dry with paper towels. Season both sides with salt and black pepper.
Heat olive oil in a large skillet over medium-high heat until shimmering.
Add the chicken breasts to the skillet and cook for 5-6 minutes on each side, or until golden brown and cooked through (internal temperature reaches 165°F). Remove the chicken from the skillet and set aside on a plate.
Reduce heat to medium. Add minced garlic to the skillet and sauté for 30 seconds until fragrant, stirring constantly to avoid burning.
Pour in the chicken broth, lemon juice, lemon zest, and dried oregano. Stir to combine and bring the mixture to a simmer.
Add the fresh baby spinach to the skillet and cook, stirring occasionally, until the spinach is wilted, about 2-3 minutes.
Return the cooked chicken breasts to the skillet, spoon some sauce and spinach over the top, and cook for an additional 2 minutes to warm through.
Remove the skillet from heat and stir in the unsalted butter until melted and the sauce is glossy.
Serve the chicken breasts topped with the lemon spinach sauce immediately.