Prepare the peaches: Split and core the peaches, then cut the halves into 1/2 to 3/4-inch wedges. Toss them in a large bowl with lemon juice, and set aside while you make the salted caramel.
Make the caramel: In a deep 4-quart ovenproof skillet (I use my braiser)* mix together the sugar and water. Turn the heat to medium-high and cook, without stirring, until the surface is bubbling evenly all over and you begin to see golden and medium brown specks in the bubbling sugar, about 5 minutes. Carefully (it will smoke noticeably) stir in the butter and continue to cook until the sauce takes on a medium brown or bronze color, about 2 to 4 minutes. Add the cream a splash at a time, stirring constantly, until evenly mixed, then cook for another three to 4 minutes. Add hot salt and stir to combine.
Remove from the heat long enough to carefully place 2/3 cup of the sauce in a bowl or container for later, then set that portion aside.
Serve the peaches: Return the pan to the stove over medium heat, and add the peaches to the remaining caramel in the pan. The cooling of the peaches will cause the caramel sauce to set a bit, but don’t worry. Just let the peaches and caramel warm and the sauce will release. Cook the peaches and caramel together, stirring frequently, until the peaches are slightly softened, about 5 minutes. Stir in the cornstarch or tapioca flour until it disappears.
Make the puree topping: Mix all the topping ingredients in a large bowl until the butter is evenly distributed and the mixture is rubbly. Sprinkle the topping over the peaches, then transfer the pan to the oven and bake until the peaches are soft, their juices seem to pool around the glossy topping, and the topping is light, nutty brown, about 20 minutes. Let cool for 10 minutes, if you can handle it, before serving.
Serving: Scoop the crisp into bowls and top with ice cream and a drizzle of the reserved caramel sauce. If your caramel sauce is cool and thick, you can heat it quickly in the microwave – check every 10 seconds, as it heats up and bubbles quickly – or in a small pan on the stove.
Do it ahead: The remaining crisp should be stored in the fridge for 5-6 days, but it would not be unprecedented. Leftover caramel sauce will keep for two weeks.
* If you don’t have a stove-top pan, just use a large, deep caramel skillet and cook the peaches, then transfer the peaches and sauce to a 4-quart or 9×13-inch baking dish. half baking pan, sprinkle the topping just before it goes into the oven.
