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, raise the heat to medium-high, and cook 5 to 7 minutes, stirring constantly, to caramelize the mixture. You'll smell a characteristic nutty-caramel odor when the butter browns, signaling that 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 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.