Place a rack in the center of the oven and preheat the oven to 425 degrees F.
Cut pockets in the chicken breasts: Place a cutting board on the counter, making sure that it is steadily in place and doesn't slide around (if it is slipping, place a damp paper towel underneath). Lay a breast on the cutting board and press your hand on top to hold it into place. Insert the tip of a sharp chef’s knife into the center of the thickest part of the breast. First, cut a slit that is 4 inches wide in the thickest part (leave the thin end uncut). Then, use the knife to make the opening deeper, cutting 3/4 of the way through the breast towards the other side. Do not cut all the way through. Repeat with the remaining breasts. With paper towels, pat the chicken dry, then sprinkle the outsides with 3/4 kosher salt.
In a large, oven-safe, non-stick skillet, heat the olive oil over medium-low heat. Add the shallot and cook until beginning to soften, about 3 minutes. Adjust the heat as needed so that the shallot softens but does not burn.
Stir in the sage, oregano, pepper, nutmeg, remaining ½ teaspoon salt, and last the garlic. Cook, stirring constantly, until fragrant, 30 seconds to 1 minute.
Add the spinach a few handfuls at a time, stirring until it wilts. Remove from the heat.
Stir in the feta, walnuts, and Parmesan.
Stuff the filling inside the breast, dividing it evenly between each.
Secure the opening with toothpicks, threading the toothpicks so that they are parallel to the counter (you'll need 1 or 2 toothpicks per breast).
With paper towels, carefully wipe out the skillet and return it to medium heat. Add the butter.
Once the butter melts, add the chicken (be careful to lower it into the pan away from you) and sear on each side for 2 to 3 minutes, until light golden.
Transfer the skillet to the oven and bake for 12 to 15 minutes, until the chicken registers 165 degrees F on an instant read thermometer. Transfer to a plate, cover and let rest 5 minutes before serving. Enjoy, with a squeeze of fresh lemon over the top.
TO STORE: Refrigerate chicken in an airtight storage container for up to 3 days.
TO REHEAT: Rewarm chicken in a baking dish in the oven at 350 degrees F.
TO FREEZE: Freeze leftovers in an airtight freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating.