In a skillet over medium heat, heat oil. Add chicken and season with salt, pepper, and oregano. Cook until browned on both sides and an instant-read thermometer inserted into the thickest part of the breast reads 165°, about 8 minutes per side. Remove from skillet and set aside.

In the same skillet over medium heat, melt butter. Stir in garlic and cook until fragrant, about 1 minute. Add cherry tomatoes and season with salt and pepper. Cook until tomatoes are beginning to burst, then add spinach and cook until spinach is beginning to wilt.

Stir in heavy cream and Parmesan and bring mixture to a simmer. Reduce heat to low and simmer until sauce is slightly reduced, about 3 minutes.

Return chicken to skillet and cook until heated through, about 5 to 7 minutes.

Serve with lemon wedges.