How to Make It
Preheat the oven to 350°. Spread the coconut out in a small baking pan and bake for 12 minutes, or until golden brown.
Lightly coat an 8-inch cake pan with nonstick cooking spray. In a small saucepan, combine 1 cup of the sugar with the water and corn syrup. Bring to a boil over moderate heat; do not stir. Cook until an amber caramel forms, 6 to 8 minutes. (Brush the side of the saucepan with a wet pastry brush if crystals form.) Working quickly, pour the caramel into the prepared cake pan tilting to coat the bottom. Immediately, sprinkle 1/4 cup of the toasted coconut over the caramel.
In a medium saucepan, warm the coconut milk and cream with the remaining toasted coconut over moderately low heat until it starts to simmer, 15 minutes. Remove from the heat and let steep for 30 minutes. Set a fine mesh sieve over a large glass measuring cup. Pour in the steeped coconut milk, pressing on the solids with a rubber spatula. You should have about 3 cups of liquid. Wipe out the saucepan.
In the same saucepan, whisk the cream of coconut with the steeped coconut milk. Warm over moderate heat until it starts to bubble and an instant read thermometer inserted into the milk reads 160°, about 10 minutes.
Meanwhile, in a medium heatproof bowl, whisk the whole eggs with the egg yolks, the remaining 2/3 cup of sugar, the vanilla extract, rum and salt. Whisk the warm coconut milk into the eggs, then strain the custard through a fine sieve set over a large glass measuring cup or bowl.
Line a roasting pan with a kitchen towel and place the prepared cake pan on the towel. Pour the strained custard into the cake pan. Add enough boiling water to the roasting pan to come halfway up the side of the cake pan. Tent the roasting pan loosely with foil and bake the flan for 55 minutes, until just set but still jiggly in the center. Remove the cake pan from the roasting pan and transfer to a rack and cool to room temperature, 1 hour. Cover with plastic wrap and refrigerate until the custard is firm, at least 2 hours or overnight.
Run a paring knife around the edge of the flan and then invert a plate over the pan. Invert the flan onto the plate, shaking gently to release. Serve immediately.