Prep the pan. Heat the oven to 325°F (163°C). Wrap a 9-inch springform pan in two layers of heavy-duty foil, covering the bottom and sides to guard against water leaks. Lightly grease the inside.
Toast the coconut. Spread shredded coconut on a baking sheet and bake for 4–6 minutes, stirring once, until lightly golden. Set aside to cool.
Make the crust. Mix graham crumbs, toasted coconut, sugar, melted butter, and a pinch of salt until the texture resembles damp sand. Press firmly into the bottom and 1 inch up the sides of the pan. Bake for 10 minutes, then cool on a rack while you make the filling.
Beat the cream cheese. In a large bowl, beat cream cheese on medium speed until smooth and fluffy, about 2 minutes. Add sugar and beat another minute until no lumps remain. Scrape the bowl well.
Add eggs and flavorings. Beat in the eggs one at a time on low, just until combined. Mix in vanilla, coconut extract, salt, and cornstarch. Avoid overmixing to limit air bubbles.
Blend in coconut milk. With the mixer on low, slowly pour in the coconut milk. Mix until silky and uniform. Tap the bowl on the counter a few times to pop air pockets.
Set up the water bath. Pour the filling into the crust. Place the springform pan inside a large roasting pan. Add hot water to the roasting pan until it reaches halfway up the sides of the springform.
Bake low and slow. Bake at 325°F for 55–70 minutes, until the edges are set but the center still wobbles like gelatin. Do not bake until firm; carryover heat finishes the job.
Spread the sour cream topping. Stir sour cream, sugar, and vanilla until smooth. Gently spread over the hot cheesecake. Return to the oven for 5 minutes.
Cool gradually. Turn off the oven, crack the door, and let the cheesecake rest for 30 minutes. Remove the pan from the water bath, unwrap the foil, and cool on a rack until room temperature.
Chill to set. Cover and refrigerate at least 6 hours, preferably overnight. This step sets the texture and makes slicing clean and easy.
Garnish and serve. Run a thin knife around the edge, then release the springform ring. Top with toasted coconut, a little lime zest, or whipped cream. Slice with a hot, dry knife for neat wedges.