Prepare the peaches: Halve and pit the peaches, then cut the halves into 1/2-to-3/4-inch-thick wedges. Toss them in a large bowl with the lemon juice, and set aside while you make the salted caramel.
Make the caramel: In a 4-quart ovenproof deep skillet (I’m using my braiser)* stir together the sugar and water. Turn the heat to medium-high and cook, without stirring, until the surface is evenly bubbling all over and you begin to see golden and medium-brown spots in the bubbling sugar, about 5 minutes. Carefully (it’s going to sizzle dramatically) whisk in the butter and continue to cook until the sauce takes on an even medium-brown or copper color, about 2 to 4 minutes. Add the cream a splash at a time, whisking constantly, until it’s evenly mixed, and cook for another 3 to 4 minutes. Add flaky salt and stir to combine.
Remove from the heat just long enough to carefully ladle 2/3 cup of the sauce carefully into a spouted dish or bowl for later, and set that portion aside.
Parcook the peaches: Return the pan to the stove over medium heat, and add the peaches to the remaining caramel in the pan. The coolness of the peaches will cause the caramel sauce to seize up a little, but do not fret. Just let the peaches and caramel warm up and the sauce will loosen. Cook peaches and caramel together, stirring frequently, until the peaches have softened slightly, about 5 minutes. Stir in the cornstarch or tapioca flour until it disappears.
Make the crisp topping: Mix all of the topping ingredients in a large bowl until the butter is evenly dispersed and the mixture is rubbly. Sprinkle the topping over the peaches, then transfer pan to the oven and bake until the peaches are tender, their juices visibly bubbling around the crisp topping, and the topping is a light, nutty brown, about 20 minutes. Let it cool for 10 minutes, if you can bear it, before serving.
To serve: Scoop the crisp into bowls and top with ice cream and a drizzle of the reserved caramel sauce. If your caramel sauce has cooled and firmed up, you can warm it up quickly in the microwave — check every 10 seconds, as it warms and bubbles up quickly — or in a small saucepan on the stove.
Do ahead: Leftover crisp should keep in the fridge for 5-6 days, but it would be unprecedented. Leftover caramel sauce will keep for two weeks.
* If you don’t have a pan that goes from stove to oven, simply use a large, deep skillet for the caramel and parcooking the peaches, and transfer the peaches and sauce to a 4-quart or 9×13-inch baking dish pan for the baking portion, sprinkling the topping on just before it goes in the oven.
