Preheat oven to 250°. Line 2 baking sheets with parchment paper and draw a 5- by 10-in. rectangle on each with a pencil; turn paper over.
Make meringues: Pulse almonds in a food processor until coarsely ground. In a large bowl, beat egg whites and cream of tartar with a mixer on high speed until frothy. Gradually add 1 cup sugar (about 1 tbsp. every 10 to 15 seconds), beating until whites are glossy and hold a straight point when beater is lifted. Scrape inside of bowl. Beat in almond extract until evenly blended, then fold in ground almonds.
Attach corners of parchment to pans with a little meringue. With a long metal spatula, evenly spread half of meringue on each parchment rectangle, shaping straight sides. Bake, switching pans halfway through, until meringues are very crisp and pale golden, about 1 1/2 hours. Let stand in oven with door ajar until cool, at least 2 hours.
Meanwhile, make caramel: Peel* 2 medium peaches (wait till step 6 to peel others), pit, and finely mash in a bowl with a pastry blender. Cook 1 cup sugar in a large frying pan over medium heat, lifting and swirling pan occasionally, until sugar melts and turns golden, 6 to 8 minutes. Stir in mashed peaches (sugar may seize up) and remove from heat. Let stand about 30 minutes. If you still have sugar lumps, cook over medium-low heat, stirring often, until sugar melts again, 3 to 6 minutes. Spoon 1/2 cup peach mixture into a medium bowl, add lemon juice, and set aside.
Strain remaining peach mixture into a bowl, pushing to extract liquid, then return liquid to pan. Stir in butter and salt. Cook over medium heat, stirring, until butter melts. Stir in almond extract and set the salted caramel aside.
Peel remaining peaches, slice, and stir into fruit-lemon juice mixture in bowl. In another bowl, beat cream and remaining 1 tbsp. sugar until softly thickened.
Loosen meringues from paper. Set 1 on a platter. Gently spread with half of whipped cream, leaving a border. Spoon some of sliced peaches and their liquid on top. Drizzle with a little salted caramel sauce (rewarm if needed so it's pourable). Repeat layers. Cut into wide slices and serve with more peaches and any remaining caramel sauce.
*To peel peaches, immerse them in a pan of boiling water until skins pull off easily with a knife, about 15 seconds. Remove with a slotted spoon and peel them with a knife.
Make ahead: Through step 5, up to 1 day; warm sauces in a microwave until room temperature.