Preheat the oven to 375°F (190°C). Line a baking dish with parchment paper or lightly grease it with cooking spray.
Place the chicken breasts on a cutting board. Using a sharp knife, make several deep cuts (about 3/4 of the way through) in each chicken breast, spacing them about 1 inch apart. Be careful not to cut all the way through.
In a medium bowl, combine the chopped spinach, crumbled goat cheese, and halved cherry tomatoes. Mix well to combine.
Drizzle the olive oil over the chicken breasts and season with garlic powder, onion powder, oregano, salt, and pepper.
Stuff each cut in the chicken breasts with the spinach and goat cheese mixture, pressing it in gently.
Place the stuffed chicken breasts in the prepared baking dish. Drizzle any remaining olive oil over the top.
Bake in the preheated oven for 25-30 minutes, or until the chicken is cooked through and reaches an internal temperature of 165°F (75°C).
Remove from the oven and let the chicken rest for 5 minutes before slicing. Garnish with fresh basil leaves if desired.