Heat the oven to 350°F. Set an 8- or 9-inch round cake pan inside a larger roasting pan. Bring enough water to a boil to come about halfway up the sides of the cake pan.
Add the eggs, egg yolks, condensed milk, evaporated milk, vanilla bean paste, powdered milk, and salt to a blender. Blend just until smooth, about 20 to 30 seconds. Let the mixture rest briefly so excess foam can settle.
Combine the sugar and water in a small saucepan. Cook over medium heat until the sugar dissolves, then let it boil without stirring until the syrup turns a deep golden amber. Gently swirl the pan if the color develops unevenly. Stir in the pinch of salt.
Immediately pour the hot caramel into the round cake pan. Carefully tilt the pan so the caramel coats the bottom, then let it stand for about 5 minutes. It will harden as it cools, which is normal.
For the smoothest texture, pour the custard through a fine-mesh strainer into the caramel-lined pan. Cover the pan tightly with foil.
Place the roasting pan on the oven rack. Carefully pour the hot water into the roasting pan until it reaches about halfway up the sides of the flan pan.
Bake for 40 to 50 minutes. The edges should look set while the center still has a soft, unified wobble when the pan is gently moved. Do not bake until the center is completely firm.
Remove the flan pan from the water bath and cool for 30 to 45 minutes. Cover and refrigerate for at least 4 hours, preferably overnight.
To unmold, run a thin knife around the edge. Place a rimmed serving plate over the pan and invert in one confident motion. Lift the pan slowly and let the caramel sauce run over the flan before slicing and serving cold.