Preheat the oven to 375 degrees F.
To make the filling, mix together the goat’s cheese, spinach, garlic powder, lemon zest, and red pepper flakes (if using) in a medium bowl until well combined. Set it aside.
To prepare the chicken, pat dry chicken breasts on each side using paper towels. Season with salt and pepper on all sides.
Carefully cut a horizontal slid in the thickest part of each chicken breast making sure to not cut through the back of the breast.
Fill each “pocket” with 2 tablespoons of the filling and gently press to distrubute it evenly. Secure with 2-3 toothpicks.
Heat vegetable oil in a 10 or 12-inch oven proof skillet or cast iron skillet, over medium-high heat until shimmering hot.
Sear stuffed chicken breasts until lightly browned on both sides. This takes about 4-5 minutes on each side. Transfer the now semi-cooked spinach stuffed chicken breasts onto an empty plate and cover it loosely with aluminum foil.
Melt the butter in the now-empty skillet. Add the shallot and cook, stirring constantly, for a minute. Pour in the wine. Using a wooden spoon, scrape up any browned bits from the bottom of the skillet. Let it simmer for 2-3 minutes or until it is reduced by half. Stir in the lemon juice.
Add the stuffed chicken back to the skillet. Transfer to the preheated oven and bake until a thermometer inserted in the thickest part of one of the chicken breasts registers 165 degrees Fahrenheit, 18-20 minutes.
When ready to serve, carefully remove the toothpicks from the chicken breasts. Drizzle them with the sauce at the bottom of the pan.
If preferred, finish it off with a drizzle of balsamic glaze and sprinkle with fresh basil. Serve right away.