Line 2 baking sheets with parchment paper. Heat the vegetable oil in a pot over medium-high heat. Add the popcorn kernels; cover and cook, shaking the pan, until the popping subsides. Remove from the heat and let any remaining kernels pop for 30 more seconds. Divide the popcorn between the prepared baking sheets, discarding any unpopped kernels. Scatter the apple chips and almonds on top.
Make the caramel: Bring the apple juice, brown sugar and corn syrup to a boil in a large heavy-bottomed saucepan over medium-high heat. Cook without stirring until the mixture is deep amber and a candy thermometer registers 280 degrees F, about 30 minutes (the caramel will bubble vigorously toward the end of the cooking time).
Stir the butter into the caramel until melted. Add 2 teaspoons salt and the apple pie spice. Add the applejack and cayenne, if using, and cook until slightly thickened, about 1 minute. Remove from the heat. Pour the hot caramel over the popcorn mixture and carefully toss with two rubber spatulas. Let cool slightly, then separate the mixture into clusters while still warm. Let cool completely.