Preheat oven to 350 F. Place 1 cup of sugar in a small saucepan over medium heat. Using a heat-proof spoon, continuously stir sugar as it begins to clump, then melt into a golden-brown sauce. Remove from heat immediately and carefully pour into a 9-inch flan (or pie) dish. Protect your hands with a towel or oven mitts. Quickly tilt dish to evenly coat the bottom and up the sides before the syrup hardens (it will become a hard, sugarlike candy).
With a whisk, beat eggs, 1/2 cup sugar, and vanilla in a medium bowl until blended but not foamy. Mix evaporated milk and coconut milk in a medium saucepan. Heat until steaming but not bubbling. Very slowly whisk hot milk into the egg mixture to avoid curdling.
Place the sugar-coated dish in a roasting pan/casserole dish large enough that it can hold the dish without the sides touching the edges of the pan. Evenly pour the egg mixture over the hardened sugar. Move to preheated oven, then pour hot water into the bottom of the roasting pan, avoiding the flan dish, until water reaches halfway up the side of the flan dish. Cover roasting pan with aluminum foil.
Bake 40 to 50 minutes, checking at 30 minutes and every 10 minutes after. Insert knife about 1 inch from the center of the custard. If knife is clean when pulled out, the custard is done. If any custard clings to the blade, bake a few minutes longer and test again. Flan should still be wiggly in the center.
When done, remove from oven and cool at room temperature in the water bath. Once cooled, gently loosen edges with the tip of the knife. Invert onto a large serving dish. Be sure to use a dish deep enough to contain the sauce that’ll drip down the sides. Cut into 8 slices, serve with sauce spooned over top, and sprinkle with the shaved coconut.
Serve warm or cold. Store in refrigerator.
Per serving: 384 calories, 21 grams fat (16 grams saturated fat), 46 grams total carbohydrates, 2 grams fiber, 8 grams protein.