Unfold pastry and press on fold lines to flatten. Cut pastry into 4 squares. Cut off scant 1/2-inch-wide strip from each side of each square (for total of 16 strips). Pierce pastry squares all over with fork. Brush edges with egg glaze. Place dough strips atop edges of squares, trimming to fit. Press gently to adhere. Brush strips with glaze. Transfer to cookie sheet. (Can be made 1 day ahead. Cover; freeze. Thaw 10 minutes before continuing.)
Preheat oven to 400F. Mix sugar, lemon peel, cinnamon and nutmeg in small bowl. Sprinkle 2 teaspoons mixture onto center of each pastry square. In medium bowl, toss nectarines with 2 tablespoons sugar mixture. Arrange nectarines diagonally in pastry squares. Brush nectarines with honey. Bake until pastries are golden, about 25 minutes. Transfer to plates. Serve warm with ice cream.