Preheat oven to 350ºF.
In a medium bowl, lightly beat the eggs, then gently stir in the condensed milk, coconut milk and vanilla, making sure they are thoroughly combined. Use a whisk or wooden spoon for this, not an electric mixer. Set aside.
Put the sugar and a tablespoon of water into a medium saucepan. Melt the sugar over a medium flame, stirring continuously with a wooden spoon or spatula until it melts. It will start to brown or caramelize as it melts, and that’s okay. When it is completely melted and dark golden brown, carefully pour it into the bottom of your pie plate (this stuff is really hot, so do be careful). Gently swirl it around in the pie plate to evenly coat the bottom and the sides. Let it cool a moment to harden slightly.
Pour the egg/milk mixture into the pie plate. Preferably strain it through a sieve, in case there are bits of anything that didn’t totally incorporate into the mixture. Place the pie plate into a roasting pan or large baking dish. Carefully pour boiling water into the roasting pan, avoiding splashing water into the flan dish, until it comes about halfway up the sides of the pie plate. If you don’t have enough boiling water, add hot tap water.
Transfer the entire operation to the oven and bake for 45 to 55 minutes (topping up the water bath, if needed), until the center of the flan is set and a toothpick inserted in the middle comes out clean.
Remove the pie plate from the water bath and cool on a rack. Then cover and refrigerate for at least 4 hours or overnight.
To serve, run a sharp knife around the edge and gently invert it on a large rimmed serving plate; we placed our serving plate face down over the pie plate, then turned everything together. (see Kitchen Notes about our unwarranted worries doing this). Slice into wedges and serve.