Crack the eggs into a large mixing bowl, add a pinch of salt on top and whisk them well. Mix only until the eggs are evenly combined and the salt has dissolved; you do not want the mixture to become very foamy. Set the bowl aside; this way, the custard will keep a nice deep yellow color once baked.
Place 1 cup (200 g) of sugar in a heavy-bottomed saucepan and set it over low heat. Stir continuously with a wooden spoon until the sugar melts completely and turns a pleasant golden color. Do not let the sugar burn; lift the saucepan off the heat from time to time, stirring vigorously so the caramel stays smooth and fluid without a bitter taste.
Choose a tall saucepan or pot of about 3–4 liters (3–4 quarts) so the custard bakes up high and looks impressive when turned out. Make sure you also have a large round serving plate ready that will fit over the pot when you invert the crème caramel at the end.
Tilt and rotate the pot so the hot caramel coats the bottom and sides evenly, forming a thin layer of caramel all around. Set the pot aside and let the caramel cool and harden completely.
Add the remaining 1 cup (200 g) sugar and the vanilla to the beaten eggs, then whisk until the sugar is fully dissolved. Pour in the milk over the egg and vanilla mixture and mix for about 1 minute, just until smooth and homogeneous. Carefully pour this custard base into the pot lined with hardened caramel.
Preheat the oven to 395°F (200°C). Place the pot with the custard into a larger roasting pan or another deep baking dish and pour cold water around it, so the water level comes about halfway up the sides of the custard pot. Slide everything into the oven, then immediately reduce the temperature to 320–340°F (160–170°C) to bake the custard gently in a water bath (bain-marie).
After about 10 minutes of baking, pour roughly 3 tablespoons (about 50 ml) of cold water into the roasting pan and repeat this every 10 minutes. This keeps the water temperature just below boiling, so the eggs set slowly and evenly. The custard will bake for about 40–50 minutes in total, until it is set around the edges but still slightly jiggly in the center, with a silky texture and no large air pockets.
Remove the roasting pan from the oven and leave the pot in the hot water for another 10 minutes so the custard transitions gradually from oven heat. Then lift the pot out of the water and let the custard cool completely at room temperature. Do not place it straight into the fridge while hot, or it may sink and lose its shape. Once fully cooled, refrigerate the pot until the next day; this resting time helps the custard and caramel syrup set beautifully together.
The next day, run a thin knife around the edge of the custard to loosen it from the pot. Place the round serving plate on top, hold both firmly and invert in one confident motion; be careful, as there will be plenty of caramel syrup that can spill. Slice the crème caramel into wedges, like a cake, keep it chilled and serve it cold, plain or topped with a dollop of lightly sweetened vanilla whipped cream. If you like, you can sprinkle raisins soaked in rum or lightly toasted walnut halves over the hot caramel at the beginning; they will stick to the caramel and create a delicious crunchy topping when unmolded.