To prevent clumps in the custard, use a freshly opened package of brown sugar.
Make the custards: Melt butter in a large saucepan over medium heat. Add brown sugar and cook, stirring, until dissolved. Remove from heat and stir in 1 cup cream and the vanilla.
Add remaining 1 1/2 cups cream, the milk, and salt to the brown-sugar mixture. Bring to a simmer over medium-high heat. Pour about 1/3 of the cream mixture into the egg yolks, whisking constantly. Still whisking, add to the remaining cream mixture in pan. Pour through a fine sieve into a bowl. Place plastic wrap directly on surface to prevent a skin from forming. Refrigerate for at least 2 hours (or up to 2 days).
Preheat oven to 325 degrees. Divide custard among six 6-ounce heatproof glasses or custard cups. Place in a roasting pan. Add enough hot water to pan to reach 2/3 of the way up sides of glasses. Cover pan with foil, and prick all over with a fork. Bake for 25 minutes. Carefully loosen a corner of the foil. Continue to bake until edges of custard are set but centers are still wobbly, 10 to 15 minutes more. Transfer glasses to a wire rack to cool. Cover, and refrigerate overnight.
Make the caramel: Combine granulated sugar, water, and corn syrup in a small saucepan. Cook over medium-high heat, stirring, until sugar dissolves. Raise heat to high, and cook until mixture turns medium amber, about 3 minutes (swirl pan gently to distribute color). Carefully stir in apple cider (it will spatter). Let cool. Drizzle caramel over custards, or serve it on the side.