First, make the apple butter: Tie the ginger, allspice, cinnamon sticks and star anise in cheesecloth for easy removal from the pot. Combine the apples, vinegar, water and spice sachet in a large heavy-bottom pot over high heat. Bring to a simmer and reduce heat to medium-low. Cook, stirring occasionally, until the apples are completely softened and the liquid has reduced by half, about 30–40 minutes. Remove from the heat and let cool slightly. Remove the spices and set them aside.

Process the apples either with a food mill, blender or food processor until smooth. Place the blended apples, spice sachet and both sugars in the pot. Cook on medium-low heat, stirring occasionally, until the mixture is thick, glossy and a deep golden brown (2–2 ½ hours). Once it becomes thicker, stir more frequently. Add the salt.

Preheat oven to 300°F. Spread the apple mixture into a large baking dish. Bake for about 3–3½ hours. Stir every 30 minutes. Cool. (This can be made in advance and refrigerated for up to 2 weeks.)

To make the frangipane: Preheat oven to 325°F. Spread the semolina flour evenly on a lined sheet pan and toast flour in the oven until it is fragrant, about 10–15 minutes. Cool to room temperature. Cream together the butter and sugars in a stand mixer fitted with a paddle attachment until fluffy. Scrape well. Add the eggs and vanilla slowly. Scrape well again. Add both flours and mix until just combined.

For final assembly, first spray 8 mini tart pans with nonstick spray. Press frangipane into the tart pans until you get a ¼ inch thickness on the bottom and up the sides. Spoon apple butter into the middle. Leave about ¼ inch of space at the top. Top the tarts with additional frangipane. Spread until smooth; seal and trim the edges. Bake at 325°F for 20–25 minutes, or until evenly golden.

Let cool slightly. Top with powdered sugar and serve either warm or at room temperature.