In a large skillet over medium heat, heat oil. Sprinkle chicken all over with salt, pepper, and oregano. Cook, turning halfway through, until browned on both sides and an instant-read thermometer inserted into thickest part registers 165°, about 8 minutes per side. Transfer to a plate.

In same skillet over medium heat, melt butter. Stir in garlic and cook, stirring, until fragrant, about 1 minute. Add tomatoes; season with salt and pepper. Cook, stirring, until tomatoes are beginning to burst, about 5 minutes. Add spinach and cook, stirring, until spinach is beginning to wilt, 2 to 3 minutes more.

Stir in cream and Parmesan and bring to a simmer. Reduce heat to low and continue to simmer, stirring occasionally, until sauce is slightly reduced, about 3 minutes. Return chicken to skillet and cook, stirring occasionally, until heated through, 5 to 7 minutes.

Divide chicken among plates. Spoon sauce over. Serve with lemon wedges alongside.