Make the ice cream cake: Preheat the oven to 400˚. Put the puff pastry halves on a baking sheet, leaving a little space between each piece. Top with another baking sheet to keep the pastry from puffing up too much. Bake until golden, about 30 minutes.
Combine the sugar and cinnamon in a small bowl. Remove the top baking sheet and brush the pastry with half the melted butter. Sprinkle with half the cinnamon sugar, then flip the pastry; brush with the remaining butter and sprinkle with the remaining cinnamon sugar. Let cool.
Coat a 9-by-5-inch loaf pan with cooking spray and line with plastic wrap, leaving a 6-inch overhang on the two short ends. Trim the puff pastry pieces if necessary so they fit snugly in the pan; set aside.
Scoop half the ice cream and sorbet into the pan, alternating flavors. Use a piece of plastic wrap to pack in the ice cream evenly. Spread with half the jam and top with 1 piece of puff pastry. Repeat with the remaining ice cream and sorbet, jam and puff pastry. Cover with the overhanging plastic wrap and freeze until firm, at least 6 hours or overnight.
Make the meringue: Beat the egg whites and cream of tartar in a large bowl with a mixer on medium-high speed until foamy, 30 seconds to 1 minute. Gradually beat in the sugar. Increase the mixer speed to high and beat until stiff, shiny peaks form, 4 to 5 minutes. Add the freeze-dried strawberries and beat until combined, about 1 minute.
Uncover the ice cream cake and invert onto a foil-lined baking sheet. Let stand until the ice cream cake slips out of the pan easily; unwrap. Spread one-quarter of the meringue over the top and sides. Transfer the remaining meringue to a large piping bag fitted with a 1/2-inch star tip. Pipe over the top and sides of the cake. Freeze until firm, at least 4 hours and up to 2 days.
Brown the meringue with a kitchen torch (or bake at 500˚ until golden, 3 to 5 minutes). Use two large spatulas to transfer the baked Alaska to a platter and slice.