For the cherry-basil filling:
For the almond crisp:
Pour the water and the sugar into a medium pot. Turn the heat to medium and stir until the sugar is dissolved. Add the basil. Simmer until the syrup is thickened and has reduced by 2/3. Discard the basil and let the syrup cool. (You can make this syrup several days in advance, and store it in the fridge.)
In a mixing bowl, toss the cooled basil syrup with the pitted cherries. Set the bowl aside.
Empty the slivered almonds into the bowl of a food processor; whiz for about 30 seconds, until the almonds are reduced to dusty crumbles.
Put the almond crumbles, flour, salt, and butter into a large bowl. Use your fingers to make small "pebbles." Add the sugar and keep smushing with your fingers until the mix resembles wet sand.
Empty the basil-cherries into a medium-size casserole or baking dish. Top the cherries with the crisp topping. Move the dish to the oven for about 25 minutes, or until the crisp is a shade darker and the cherries are bubbling. Serve warm, with vanilla ice cream, if you like.