Prepare an ice bath by nesting a smaller bowl inside a larger bowl filled with ice water. You can also use a zip-top freezer bag set inside a bowl of ice water.Warm cream in a small saucepan until warm to the touch (do not boil). Reduce heat to keep warm while you prepare the caramel.Place sugar in an even layer in the bottom of a large heavy saucepan set over medium heat. Keep an eye on it, but once the sugar starts to liquefy around the edges, use a heat-proof rubber spatula to carefully drag the liquid toward the center. Continue to drag the liquid edges toward the center, doing so slowly, until all the sugar is melted and golden brown. If at any point the color begins to get too dark and you still have granulated sugar, lower the heat and gently stir to dissolve the remaining sugar. The final caramel should have a faintly burnt smell and deep amber color. Watch this carefully as caramel can go from not quite done to burnt in a matter of seconds (see more tips for making a dry caramel here).Remove caramel from heat and slowly pour in warm cream, whisking vigorously the entire time (caramel will bubble up so be careful as it is VERY hot). Continue to whisk, scraping the sides and corners of the pan, until no hard chunks remain. If necessary, return to medium-low heat and whisk until any remaining chunks are dissolved. Add butter and chocolate and whisk until smooth and chocolate is melted. Whisk in cinnamon, salt, and cayenne, followed by milk.Return mixture to medium heat until it just begins to steam.In a small bowl, whisk egg yolks. Slowly ladle in some of the warm chocolate mixture, about 1/4 cup at a time, until about half of the chocolate mixture has been incorporated and yolks are warm to the touch. You want to do this gradually; doing so will temper the egg yolks rather than cook them.Pour entire yolk mixture back into the saucepan and return to medium heat, stirring constantly and scraping the bottom of the pan as you do, until the mixture thickens slightly and coats the back of a spoon, about 5 to 7 minutes (it will measure approximately 170ºF on a thermometer). Do not let it to boil.Stir in vanilla extract. Pour through a fine mesh sieve (to catch any undesirable chunks) into prepared ice bath, and let sit for 15 to 20 minutes or until cooled to room temperature. Cover tightly with plastic wrap and refrigerate for at least 3 hours or preferably overnight.When custard base is completely chilled, churn ice cream according to manufacturer’s instructions until the ice cream is the consistency of soft serve. Spoon a into a freezer safe container and freeze overnight until firm.