Special equipment: 9-inch springform pan
For the Mango Cheesecake: Preheat the oven to 350 degrees F.
Wrap the outside of a 9-inch springform pan with 2 3/4-inch-high sides with 2 layers of heavy-duty foil. Finely grind the biscotti in a food processor. Add the melted butter and process until the crumbs are moistened. Press the crumb mixture over the bottom (not the sides) of the prepared pan. Bake until the crust is golden, about 15 minutes. Cool the crust completely on a cooling rack.
Blend the cream cheese and ricotta in a food processor. Add the mango puree, eggs, and sugar and pulse until the mixture is smooth.
Pour the mango mixture over the crust in the pan. Place the springform pan in a large roasting pan. Pour enough hot water into the roasting pan to come halfway up the sides of the springform pan. Bake until the cheesecake is firm and moves slightly when the pan is gently shaken, about 1 hour 30 minutes. Transfer the cake to a cooling rack to cool for 30 minutes. Place in the refrigerator and cool completely, at least 8 hours and up to 2 days.
For the Basil Lemon Syrup: Place the sugar, water, and lemon juice in a small saucepan. Bring the mixture to a simmer over medium heat and cook until the sugar is dissolved. Cool the syrup completely. In a food processor combine the basil and the cooled syrup. Pulse until the herbs are finely chopped. Strain the mixture through a fine mesh sieve.
Slice the cheesecake and place on a serving plate. Drizzle with the syrup and serve.