Preheat the oven to 350 degrees F.
To make the crostini, cut the baguettes into 1/4-inch thick slices. Using a pastry brush, brush the baguette slices with olive oil. Place on 2 foil-lined baking sheets and bake until crisp and golden, about 12 to 15 minutes. Reserve 25 of the crostini slices for the cheesecake. Cool the remaining crostini, then store in an airtight container until ready to serve.
To make the cheesecake crust, put the 25 crostini slices in a food processor and run the machine until crostini are like bread crumbs, about 1 minute. Add the melted butter and pulse a few times to combine. Press the bread crumbs into the bottom of the cheesecake pan. Bake until golden, about 10 minutes. Remove from the oven and let cool.
To make the cheesecake, wrap the outside of the cheesecake pan with 2 layers of heavy-duty foil. Place the ricotta, cream cheese, and goat cheese in a food processor. Pulse to mix. Add the eggs, sugar, and salt and pulse a few times until well mixed. Using a rubber spatula, stir in the red pepper, basil, and thyme.
Pour the cheese mixture into the cheesecake pan. Place the cheesecake pan in a roasting pan. Pour enough hot water in the roasting pan to come halfway up the sides of the cheesecake pan. Bake until the cheesecake is golden at the edges and the center of the cake moves slightly when the pan is gently shaken, about 1 hour (the cake will become firm when it is cold.)
Transfer the cake to a wire rack to cool for 1 hour. Refrigerate until the cheesecake is cold, at least 3 hours and up to 2 days.
To serve, combine the jam and the hot water in a small bowl and stir until the jam is liquefied. Remove the cheesecake from the springform pan by running a knife around the inside of the rim and unlatching the pan's ring. Then, run a long metal spatula under the cheesecake and gently slide onto a serving plate. Drizzle the jam mixture over the top of the cheesecake, encouraging some of the jam to run down the sides. Serve with the reserved crostini alongside.
Recipe courtesy of Giada De Laurentiis