Make the filling:
Heat the olive oil in a large skillet over medium to medium low heat. Add the shallots and cook slowly until softened and aromatic, 8 to 10 minutes. Increase the heat to medium and add the spinach (in batches, if necessary) and wine. Cook, stirring frequently, until the spinach is wilted and all the liquid has evaporated, 3 to 4 minutes. Remove from the heat and stir in the goat cheese, salt, and pepper. Let cool to room temperature.
Stuff and cook the chicken:
If the chicken breasts have tenderloins, remove them. Trim, rinse, and pat the breasts dry.
Make a pocket on the thicker side of each breast. Using a sharp boning or utility knife, cut into the breast about 1/2 inch from one end. Create a pocket, slicing to within about 1/4 inch of the other side.
Stuff each breast with about one-quarter of the filling, distributing it evenly throughout the pocket and to the ends. Press on the top of each breast to close the pocket.
Line up three wide shallow dishes. Fill the first with the flour. In the second, whisk the eggs. In the third, toss the breadcrumbs with 1/2 tsp. salt and 1/2 tsp. pepper.
Season the breasts generously on both sides with salt and pepper. Dredge one breast well in the flour, shaking off any excess. Dip it into the eggs, turning to coat evenly, and then dredge it in the breadcrumbs, pressing to make the crumbs adhere evenly. Gently shake off any excess. Set on a plate and repeat with the other breasts. Refrigerate for at least 5 minutes and up to 3 hours to let the breading set. Discard any leftover flour, egg, or crumbs.
Heat the oven to 350°F. Heat the oil in a heavy 10-inch nonstick skillet over medium-high heat. When the oil is very hot, carefully add two of the breasts to the pan and cook until golden brown, about 3 minutes per side. If the oil seems to get too hot, reduce the heat to medium. Transfer the breasts to a baking sheet. Repeat with the other two breasts.
Bake until the chicken and filling reach 165°F on an instant-read thermometer, about 15 minutes. Serve immediately.
The filling can be made a day ahead and refrigerated, covered.
Serve with roasted, herbed red potatoes.