01 - Preheat oven to 350°F (175°C). Line a 12-cup muffin tin with paper liners.
02 - In a bowl, whisk together flour, granulated sugar, baking powder, baking soda, salt, and chai spices. In a separate bowl, combine vegetable oil, whole milk, egg, and vanilla extract. Gradually fold the wet ingredients into the dry mixture until just incorporated.
03 - Beat softened cream cheese and granulated sugar together until smooth. Incorporate egg and vanilla extract and mix until creamy.
04 - Spoon 1 heaping tablespoon of chai cake batter into each prepared muffin cup. Add 1 tablespoon of cheesecake mixture on top, then another layer of chai cake batter, filling cups approximately three-quarters full.
05 - Bake muffins for 20 to 22 minutes, or until a toothpick inserted into the cake portion comes out clean. Transfer to a wire rack and cool completely.
06 - In a skillet over medium heat, melt unsalted butter. Sauté diced apples with brown sugar, ground cinnamon, and lemon juice until soft and caramelized, about 5 to 7 minutes. Let cool.
07 - Once muffins have cooled, slice each horizontally into two even rounds.
08 - For each wooden or bamboo skewer, alternate layers of muffin slice, spoonful of apple filling, and a drizzle of caramel sauce. Repeat to assemble two or three stacked segments per skewer. Finish with an additional drizzle of caramel and a sprinkle of chai spice if desired.
09 - Serve skewers immediately, or chill assembled skewers for 15 minutes to set layers for a firmer texture.