Preheat the oven to 400 degrees F.
In a medium-size saute pan, heat the olive oil over medium-high heat. Add the onions and cook for 5 minutes. Turn the heat down to medium and add the garlic. Cook for 5 minutes, stirring frequently. Don't let the onions and garlic brown. Next, add the chopped sage and salt and pepper to taste, and cook until the onions are translucent, 5 to 10 minutes. Add the apricots and cook until heated through, 5 minutes. Remove the pan from the heat and let cool for 10 minutes. Thoroughly mix in the goat cheese and set aside. (This can be made up to 24 hours in advance and refrigerated. Bring to room temperature before stuffing the chicken breasts.)
Place each chicken breast between 2 pieces of plastic wrap. With a kitchen mallet, a cleaver, or a heavy tin can, such as a can of tomatoes, pound the breasts until they are between 1/8 and 1/4-inch thick. (They will not roll easily if they are too thick.) Season with salt and pepper.
Spread about 1/4 cup of the stuffing down the center of each chicken breast. Fold up the short ends of the chicken breast, and then roll one long side jelly-roll-fashion to the other side. Tie the ends with string to secure them. Place the egg in a shallow bowl and the panko crumbs in another shallow bowl. Dip the chicken in the egg, and then roll in the panko crumbs until completely coated. In an ovenproof saute pan, melt 4 tablespoons of the butter over medium-high heat. Place the chicken in the pan and brown on all sides, about 8 minutes. Then put the pan in the oven and bake until the chicken is a deep golden color and firm when pressed, 10 to 12 minutes.
After the chicken breasts come out of the oven, remove the strings and put them on a warmed platter; tent with foil. Next, place the pan over high heat and deglaze it with the wine. Boil until reduced to about 1 tablespoon, 1 to 2 minutes, and then add the chicken stock. Boil until reduced by half, 3 to 5 minutes, and add salt and pepper to taste. Remove from the heat and swirl in the remaining 1 tablespoon butter.
Cut each breast in half crosswise and place one half on its side on the plate. Lean the second half against the first, facing down. Alternatively, slice breast into even slices. Drizzle with the sauce, garnish with sage leaves, and serve immediately.