Preheat the oven to 325°F.
Combine 1/4 cup of water and the cream of tartar in a small saucepan. Pour the sugar into the center of the saucepan so it does not stick to the sides. Cook, without stirring, over medium-high heat until the sugar starts to caramelize and turn dark brown, 6 to 8 minutes. Pour the liquid into an 8 inch round cake pan and rotate to coat evenly.
In a medium saucepan over medium-high heat, combine the milk, condensed milk, hazelnut extract (if using plain milk), and salt, using a spatula to gently stir. Heat until hot but not boiling. Remove from the heat.
Crack the eggs into a large mixing bowl. Using a silicone spatula, slowly break the yolks and stir the eggs gently until combined. Avoid whipping, whisking, or aggressively folding the eggs, or you will incorporate air. While mixing the eggs, slowly add ½ cup of the hot milk mixture to the eggs. Continue to stir the eggs and add another ½ cup of the hot milk. Repeat until all the milk is incorporated into the eggs. Pass this custard base through a fine-mesh sieve over a medium jug or bowl with a spout to remove any lumps.
Hold the flat end of the spatula right over the layer of caramel in the prepared baking pan. Slowly pour the custard base over the flat surface of the spatula, to prevent disturbing the layer of caramel. Cover the top of the pan with a double layer of foil.
Fill a kettle with water and bring it a boil. Place a deep, wide baking dish or pan large enough to accommodate the round baking pan in the oven. Place a circular wire rack inside the dish or make a thick, 8 in ring with aluminum foil to prevent the baking pan from touching the base of the baking dish.
Carefully place the prepared baking pan over the rack or ring in the center of the larger dish. Pour boiling water from the kettle into the space between the baking pan and the large dish and fill it to about a ½ in from the top of the baking pan. Cook the custard in the oven until it is almost set and the center jiggles slightly, 45 to 50 minutes. Carefully remove the baking pan from the oven, place it on a wire rack, and let it cool completely to room temperature. Refrigerate the flan overnight to set.
The next day, remove the foil cover. Run a small, sharp knife around the inside edge of the flan. Place a large serving plate over the baking pan. Holding them tightly, flip the pan and the plate and tap gently to release the flan onto the plate. Scrape any extra caramel liquid over the flan. Serve chilled or cold. Refrigerate any leftovers in an airtight container for up to 3 to 4 days.

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