The caramelized liquid sugar poured into the bottom of a ramekin becomes a hard candy-like top for this silky baked custard. A warm-oven water bath transforms the liquid creme anglaise into a creamy set pudding.
Preheat the oven to 325 degrees F with a rack in the center.
For the caramel: add the sugar and 3 tablespoons water to a medium saucepan over medium-high heat. Swirl the pan to wet all the sugar, but do not stir or whisk. Cook until the sugar melts and begins to bubble, swirling occasionally. It will eventually caramelize and turn golden yellow, 3 to 4 minutes.
Continue to cook until it turns an amber color, 3 to 4 minutes. Divide the caramel between 8 ramekins, coating the bottoms and partway up the sides. Set aside to cool completely and let the caramel set.
For the custard: Heat the cream, milk and vanilla in a medium saucepan over medium heat until slightly simmering. Remove from the heat.
In a large bowl, whisk together the eggs, egg yolks, sugar and salt. Temper the mixture by pouring a small amount of the hot liquid into the eggs while whisking continuously. Continue to stream the milk into eggs until all the liquid has been added and thoroughly mixed. Strain through a fine-mesh sieve to ensure a perfectly smooth custard, removing any lumps or bits of cooked egg. Divide the custard base evenly among your ramekins, pouring it over the caramel.
Put the ramekins in a large casserole dish or roasting pan. Pull out the center oven rack and place the pan on the rack. Very slowly and carefully, pour hot water into the bottom of the roasting pan until it comes about 1 inch up the sides of the ramekins. Avoid splashing water into the custards. Carefully slide in the rack and bake until the centers of custards jiggle when the ramekins are jostled, 40 to 45 minutes. If the custard is still liquid in the center and sloshes when you gently shake it, bake for an additional 2 to 3 minutes. Check often so the custards don't overbake.
Carefully remove the ramekins from the water bath with tongs and allow to cool on the countertop before transferring to the refrigerator to cool completely.
For the coconut tuiles: Toast 1/2 cup of the shredded coconut on a baking sheet until light golden brown, about 10 minutes. Stir after 5 minutes to ensure even coloring. Set aside to cool. Mix with the remaining 1/2 cup shredded coconut.
In the bowl of a stand mixer fitted with a paddle attachment, beat the butter on medium-high speed until light and creamy, 1 to 2 minutes. Add the powdered sugar and flour and mix until fully incorporated. Add the egg whites slowly, one at a time, and continue to beat until fully incorporated.
Line a baking sheet with a silicone mat. Put a small spoonful of batter on the mat and spread thinly with a small offset spatula. Sprinkle with coconut. Repeat with the remaining batter to make 8 tuiles. Bake until the edges of the cookies are golden brown but the centers are still pale, 10 to 12 minutes. Allow to cool for 1 minute after removing from the oven. Peel up the edge of one cookie with an offset spatula and peel it off the baking sheet. While still warm, shape each cookie by curling it around the handle of a wooden spoon if desired.
When ready to serve the creme caramels, run a small offset spatula or paring knife around the edge of each ramekin. Invert onto rimmed plates and carefully lift, allowing the excess caramel to fill the plate around the base of the custard. Garnish with a coconut tuile.