Make candied orange zest. Peel orange skin with peeler and cut into thin strips. Combine ½ cup water with ¼ cup sugar and orange peel in a medium saucepot. Bring to a simmer over low heat and continue to simmer for 20 minutes or until peel is softened. Remove from heat and let sit for 5 minutes. Lay orange peel on wire rack and let sit for 15 minutes. Toss with 2 tablespoons sugar until well coated and spread on parchment paper to dry completely.
Make candied cranberries and rosemary. Combine 2 tablespoons water with 2 tablespoons sugar in a medium saucepot. Bring to a simmer over low heat and continue to simmer for 2 minutes. Remove from heat and add cranberries, stirring for 1 minute. Remove with a slotted spoon and transfer to parchment paper. Add rosemary to simple syrup and stir for 1 minute. Remove with a slotted spoon and transfer to parchment paper. Let dry for 1 hour. Roll cranberries and rosemary in remaining ¼ cup sugar to coat. Set aside.
Make mulled wine caramel. Combine apple cider and red wine in a medium pot. Bring to a boil, reduce heat to medium, and continue to cook until liquid is reduced to ¼ cup.
Combine reduced wine with heavy cream, sugar, corn syrup, butter, and orange zest in a medium pot. Bring to a simmer over medium heat and continue to cook caramel until temperature reaches 225 degrees F. Remove from heat and stir in cinnamon, nutmeg, cloves, and allspice. Transfer to a heat safe bowl and let cool completely.
Preheat oven to 400 degrees F. Line the bottom and sides of 9-inch springform pan with parchment paper. Trim parchment paper so that it comes up 2 inches above the pan on all sides. Grease with butter.
Beat cream cheese, mascarpone, and sugar until smooth and creamy, about 2 minutes. Add eggs one at a time, fully incorporating the egg before adding the next. Add heavy cream, 1 ½ cups mulled wine caramel, salt, and vanilla, mixing until well combined. Add sifted flour and mix until smooth.
Pour cheesecake into prepared pan. Bake for 50 minutes - 1 hour or until the edges are set but the center cheesecake is still jiggly. Remove from the oven and let cool completely.
Garnish mulled wine caramel basque cheesecake with candied orange, cranberries, and rosemary. Serve with warm mulled wine caramel on the side.
Notes
The mulled wine caramel can be made up to 5 days in advance.