For the ice cream: In a medium saucepan, combine the cream, 1/2 cup of the granulated sugar, and vanilla bean. Bring just to a simmer, stirring occasionally, then remove from the heat.
In a small bowl, combine the eggs and the remaining 1/2 cup sugar and beat to a homogeneous consistency.
Strain a third of the cream mixture into the egg mixture, whisk to combine, and then immediately whisk the egg mixture back into the saucepan with the cream. Chill in an ice bath and place in the freezer for 30 minutes.
Churn in an ice cream machine according to the manufacturer's instructions.
For the crust: Combine the butter, flour and salt in a food processor. Pulse until the mixture looks like grated Parmesan. Drizzle in 2 tablespoons of the ice water, and pulse until the mixture forms a rough ball, adding the remaining 2 tablespoons ice water if needed. Remove the dough from the food processor and form into a disk. Wrap in plastic and refrigerate for at least 1 hour.
For the filling: Melt the butter in a small saucepan and cook it until it starts to brown and begins to smell like hazelnuts. Transfer to the bowl of a stand mixer equipped with a paddle. Add the sugar and beat on medium speed.
Gradually add the flour. Once the flour has combined, add the eggs, one at a time. Beat in the vanilla bean seeds.
To assemble: Preheat the oven to 350 degrees F.
Dust a clean work surface with flour. Roll out the dough to a 1/8-inch thickness, and cut into 4 rounds. Lay the rounds in four 4-inch ring molds, leaving lots of overhang on the sides. Schmear the filling into the bottom of the ring molds, and top with the cherries. Fold the overhanging dough over the cherries. Brush the dough with the egg wash. Bake until golden brown and crispy on top, 45 to 50 minutes.
For the caramel: In a small saucepan, combine the sugar, lemon juice and 2 tablespoons water. Bring the pan to medium-high heat; while doing so, be careful not to swish the pan around--it could cause the sugar to recrystallize. As the sugar mixture boils, the water will eventually evaporate and the sugar will begin to turn brown, or "caramelize." Pay close attention! Things can take a terrible turn for the worse here, and there will be no recovery. As the sugar turns gold, followed by light brown, and then a pale shade of darker brown, remove it from the heat and add the heavy cream. Be careful when you do this--it will bubble up like crazy, and getting it on you could cause a severe burn! Be sure to have a saucepan large enough to accommodate the bubbling up. Whisk until things settle down, then add the butter, 2 pats at a time, whisking until completely incorporated before adding more butter.
Spread some of the caramel onto four serving plates. Unmold the crostatas and set them on top of the caramel. Top with a scoop of ice cream.
This filling makes enough for two 9-inch tarts. It's easier to make a larger quantity, and it lasts in the fridge for a few days.