Preheat the oven to 325 degrees F (165 degrees C). Butter a 9-inch springform pan well.
Start the filling: Heat olive oil in a pan over medium-high heat. Saute peppers and onions with a large pinch of salt until onions just start to turn translucent, about 5 minutes. Stir in garlic and cook for 2 more minutes. Turn off heat and allow to cool to room temperature.
While the pepper mixture is cooling, prepare the crust: Combine crushed cracker crumbs, dried Italian herbs, and melted butter in a mixing bowl. Stir until crumbs are coated.
Transfer the buttered crumbs into the prepared springform pan and press down into an even layer. Set aside until needed.
Continue the filling: Combine eggs, flour, kosher salt, black pepper, cayenne pepper, and sugar in a large mixing bowl and whisk until smooth. Add the creme fraiche, cream cheese, and Parmigiano-Reggiano cheese, and mix until smooth. Stir in parsley, tarragon, thyme, and cooled pepper mixture until evenly combined.
Transfer cheese mixture into the prepared springform pan. Tap the pan gently to settle the contents and place on a baking sheet.
Bake in the lower center of the preheated oven until a skewer poked into the center comes out clean, 60 to 75 minutes. Let cool in the pan for 30 minutes. Run a knife around the edges and carefully remove the ring.
Cheesecake should be thoroughly chilled in the refrigerator before serving, about 2 hours. Cheesecake can be served at room temperature, but it will not have the classic smooth, firm cheesecake texture, and will be more like a very light cheese quiche.
