For the crust: Butter four 3-inch ring molds and set on a parchment-lined baking sheet.
In a large mixing bowl, stir together the gingersnap crumbs, melted butter, granulated sugar and salt. Press the crumb mixture into the bottom and halfway up the sides of the prepared ring molds.
For the filling: Preheat the oven to 350 degrees F. In a stand mixer fitted with the paddle attachment, beat the cream cheese and goat cheese until light and fluffy. Add the creme fraiche and beat to combine. Add the eggs, one at a time, beating after each addition until thoroughly combined. Beat in the sugar and vanilla until just combined.
Pour the filling into the prepared crusts. Bake until set, 15 to 20 minutes, rotating the baking sheet halfway through. If the cheesecakes start to color, tent the baking sheet with aluminum foil.
Remove the cheesecakes from the oven and let cool completely before serving, about 1 hour. (For best results, refrigerate overnight to ensure a proper set.)
For the tuiles: Beat the brown sugar, corn syrup, butter and salt together with an electric mixer until light and fluffy. Add the egg; beat until smooth. Mix in the flour until combined. Gently fold in the almonds. Refrigerate for 30 minutes.
Line two baking sheets with silicone baking mats. Drop heaping teaspoonfuls of the cookie dough, spaced about 3 inches apart, onto the mats. Bake until golden and crisp, about 8 minutes. Let cool slightly. When just cool enough to touch, lift the tuiles off the baking mat with a small offset spatula and gently wrap them around a rolling pin to form a curved shape. Leave on the rolling pin until set, less than a minute. Carefully remove the tuiles from the rolling pin and place on a plate. (Makes about 10 tuiles.)
For the chutney: In a small saucepan, combine the blueberries, peaches, ginger, sugar, vinegar, and water to cover. Bring to a boil over high heat, stirring often. Reduce the heat and simmer until thickened, 30 to 40 minutes.
To serve, unmold the cheesecakes onto serving plates, spoon some chutney on top and arrange a tuile on the side.