Preheat the oven to 350 degrees F.

Combine the Parmesan, bread crumbs, and butter until thoroughly blended, and press the mixture into the bottom of a 9-inch springform pan.

Heat the oil in a large skillet over high heat. Add the onions and the green and red bell peppers and sauté, stirring and shaking the skillet, for 2 minutes. Stir in the mushrooms and the salt and pepper, sauté for 1 to 2 minutes, remove from the heat.

Using an electric mixer beat the cream cheese with the eggs in a large bowl until very thick and frothy, for about 5 minutes. Beat in the cream, Gouda, sautéed vegetables, and crabmeat. Mix until thoroughly incorporated and creamy, about 2 minutes.

Pour the filling over the crust in the springform pan and bake until firm, about 1 hour and 15 or 20 minutes. Remove from the oven and allow to cool to room temperature. If you refrigerate the cake, allow it to come to room temperature, for about 1 hour before serving.

While the cheesecake is cooling, prepare the Green Onion Coulis.

To serve, cut the cake in wedges with a warm knife and serve each wedge with about 3 tablespoons of the coulis.

Note: To make homemade breadcrumbs, remove the crusts from a dense, homemade-type white bread, and process 3 or 4 slices at a time in a food processor or blender to form crumbs. Store in an airtight container at room temperature for up to 3 days or store in the freezer for up to 2 months.