Preheat the oven to 425 degrees . Heat 1 tablespoon EVOO, 1 turn of the pan, in a small skillet over medium heat. Add the onion and two-thirds of the garlic and cook until softened, 3 to 4 minutes. Transfer to a bowl to cool, then add the spinach and season with salt, pepper and nutmeg. Melt the butter in the same pan, turn off the heat and add the panko, oregano and remaining garlic; toss.
Drizzle a baking dish with the remaining 1 tablespoon EVOO. Split each chicken breast across (but not all the way through) and open like a book. Stuff with the spinach and feta, letting the stuffing overflow at the edges. Place the chicken pieces in the baking dish and top with the breadcrumbs. Bake until the juices run clear, about 20 minutes.