Place an oven rack in the center of the oven. Preheat the oven to 350 degrees F. Butter the bottom and sides of a 4 1/2-inch diameter springform pan. Line the bottom of the pan with a piece of parchment paper. Wrap the outside of the pan with a piece of heavy-duty foil.
Place the ricotta cheese, cream cheese, and goat cheese in a food processor. Pulse
until mixed. Add the sugar, egg, egg yolk, and salt and blend until smooth. Add the basil and pulse until incorporated. Pour the cheese mixture into the prepared pan. Place the pan in an 8-inch by 8-inch square baking dish. Pour enough hot water in the baking dish to come halfway up the sides of the springform pan. Bake until the cheesecake is golden at the edges and the center of the cake moves slightly when the pan is gently shaken (the cheesecake will become firm when chilled), about 50 minutes. Turn off the oven and allow the cheesecake to cool in the oven for 1 hour. Remove the springform pan from the baking dish and remove the foil. Cover the pan with plastic wrap and refrigerate the cheesecake for at least 3 hours and up to 2 days. Remove the cheesecake from the springform pan. Allow the cheesecake to come to room temperature before serving, about 30 minutes.
Using a pastry brush, brush the top of the cheesecake with extra-virgin olive oil and serve with assorted crackers.