Arrange the phyllo pastry dough sheets in a row so that the edges overlap by about 1 inch. Combine the vegetable oil and apple juice in a spray bottle and spray it over the phyllo pastry dough. Sprinkle the chopped mixed nuts on top and fold the long row in half horizontally. Spray the vegetable oil apple juice mixture over the phyllo pastry dough and fold the row in half horizontally again. Place a large, empty 6-inch can at the edge of the phyllo pastry dough row and carefully roll it up in the dough. Place the entire thing inside a greased 8-inch springform pan, transfer the springform pan to a preheated oven, and cook at 350°F for 25 minutes until golden brown. Meanwhile, add the sugar, lemon juice, lemon zest, and remaining vegetable oil apple juice mixture to a saucepan over medium heat and bring to a boil until the sugar has completely dissolved. Once the baked baklava ring has cooled, remove the empty can and carefully remove the baklava ring from the springform pan. Brush the sugar mixture over the dough and then place the baklava ring back inside the springform pan.
Combine the ingredients for the dried fruit & nut paste until a smooth mixture forms and press it into the bottom of the springform pan to create the base of the baklava cake.
Heat the raspberry jello in a saucepan over low heat until it has melted and then stir in the honey. Divide the raspberry jello mixture between 3 large bowls and fold different amounts of cream cheese into the mixture in each bowl for 3 different color gradients. Elevate one side of the springform pan so that the baklava cake is at an angle and alternate pouring different colors of the raspberry cheesecake cream into the cake pan, making sure to let each layer set before adding a new one. Refrigerate the baklava cheesecake for at least 2 hours, or preferably overnight.
Heat some honey and brush it over the chilled baklava ring and garnish the baklava cheesecake with chopped pistachios before serving.