Beat egg whites (at room temperature), cream of tartar, salt, and almond extract in a large mixing bowl until foamy. Add sugar, 1 tablespoon at a time, beating until stiff peaks form and sugar dissolves. (Do not under beat mixture.) Spoon meringue mixture into a pastry bag fitted with a star tip.
Draw 12 circles (3 inches in diameter) on 2 baking sheets covered with unglazed brown paper. (Do not use recycled paper.) Starting at center of each outlined circle, pipe meringue in a flat spiral fashion, using a circular motion that ends just inside each circle's outline. (This forms base of meringue cup.) Continue to pipe meringue atop outer ring of base, using a circular motion, to form 2 continuously attached and stacked meringue rings. (These form sides of meringue cup.) Repeat procedure with remaining meringue mixture.
Bake at 225° for 45 minutes. Turn oven off; cool meringue in oven at least 1 hour. (Do not open oven door. ) Remove baked meringue cups from brown paper. Use immediately or store in airtight containers.
Place 6 baked meringue cups on individual dessert dishes. Spoon 1/4 cup Peach Filling into each cup. Stack remaining baked cups, base side down, on top of filling. Spoon remaining Peach Filling evenly over top, filling cup and allowing excess to drizzle down sides. Top with fresh peach slices, and garnish plates with peach leaves, if desired.