Preheat the oven to 375°F (190°C). Lightly oil a baking sheet or leave an ovenproof skillet ready for finishing the chicken.
Make the filling: heat 1 tablespoon olive oil in a medium skillet over medium heat. Add the minced garlic and cook until fragrant, about 30 seconds. Add the spinach in batches, stirring until it wilts down, 2–3 minutes total. Remove from heat and let cool slightly.
Transfer the wilted spinach to a sieve or clean kitchen towel and press or squeeze out as much liquid as possible. Chop the squeezed spinach finely and place in a bowl.
Stir the crumbled feta, cream cheese, lemon zest, beaten egg, dried oregano, 1/8 teaspoon salt and a pinch of black pepper into the cooled spinach until evenly combined. Taste and adjust seasoning; the mixture should be slightly salty and cohesive.
Prepare the chicken: place a breast on a cutting board and, using a sharp knife, slice a horizontal pocket into the thickest side without cutting through. Repeat with remaining breasts. Season both inside and out with the remaining salt and pepper.
Spoon an even amount of the spinach-feta filling into each pocket, pressing gently so the filling sits flush. Use toothpicks to close the pockets if needed and tuck the ends so they hold during cooking.
Heat the remaining 1 tablespoon olive oil in a large ovenproof skillet over medium-high heat. When the oil is shimmering, add the stuffed breasts and sear until golden, about 2–3 minutes per side. Do not worry about fully cooking through at this point.
Transfer the skillet (or move the breasts to the prepared baking sheet) to the preheated oven and bake until the chicken reaches an internal temperature of 165°F (74°C), about 15–20 minutes depending on thickness.
Remove the chicken from the oven and let rest on a cutting board for 5 minutes. Remove any toothpicks, drizzle with a little fresh lemon juice if using, and serve with lemon wedges.