Use a 13-inch round serving plate with a rim to accommodate the caramel. Martha made this recipe on "Martha Bakes" episode 711.
Preheat oven to 325 degrees. Place a small bowl of cold water and a pastry brush near the stove top. Place a 9-inch cake pan in a large roasting pan. Bring a large kettle or pot of water to a boil.
Make the caramel: In a saucepan over medium-high heat, mix sugar and 3 tablespoons water, stirring to combine. Do not stir again. Cook, washing down the sides of the pan with a pastry brush dipped in water, or cover with a tight-fitting lid, to prevent crystals from forming, until caramel is amber, about 8 minutes, swirling pan to color evenly. Pour into cake pan, swirling to coat bottom evenly.
Make the custard: In a small saucepan, heat milk with 1/2 cup sugar over medium just until mixture starts to steam and bubble around the edges, 5 to 6 minutes (do not let it boil).
Meanwhile, whisk together eggs, yolks, remaining 1/2 cup sugar, and salt in a large bowl. Add ladle of hot milk mixture to the egg mixture and whisk to combine. (This is called tempering, and it prevents the eggs from curdling.) Add two more ladles of milk mixture, one at a time, whisking to combine after each addition. Gradually whisk in remaining milk mixture. Strain through a fine sieve into a large measuring cup or a bowl (to remove any cooked bits of egg). Stir in vanilla.
Pour custard into cake pan. Place roasting pan in oven. Add enough boiling water to pan to come halfway up the sides of the pan. Bake until custard is just set (it should tremble slightly in center when shaken), about 50 minutes.
Remove roasting pan from oven. Use tongs to carefully remove pan from hot-water bath and place on a wire rack for 30 minutes. Cover with plastic wrap and chill for at least 8 hours (or up to 3 days). To unmold, run a sharp knife around the flan, and place a rimmed serving plate upside down over the top. Invert, and gently lift pan to remove. Serve immediately.