In a large saucepan, combine the milk and cream and bring to a simmer over medium heat. Immediately turn off the heat and set aside.
In a large, heavy skillet, melt the butter over medium-high heat. Stir in the brown sugar and cook 5 to 7 minutes, stirring constantly to caramelize the mixture. When the butter browns the mixture is ready.
Whisking constantly, gradually add the butter/brown sugar mixture to the hot milk/cream mixture. If the mixture isn't smooth, blend for 20 seconds with a hand blender or pour through a fine sieve.
Put the egg yolks in a medium bowl. Whisk in about 1/2 cup of the hot milk mixture. Whisk in the cornstarch and salt until dissolved. Whisk the egg/cornstarch mixture back into the hot milk mixture in the saucepan.
Whisking constantly, cook over medium-high heat until thick and just boiling. When the mixture thickens, the whisk will leave trail marks on the bottom of the pot and the mixture will have a few large bubbles boiling up to the top. Turn off the heat and whisk in the vanilla extract.
Pour into 6 to 8 custard cups, ramekins, or mugs and chill, uncovered, at least 2 hours or overnight. Serve chilled.