For the black garlic paste: Put the oil and black garlic cloves in a food processor and pulse until it becomes a thick paste (about a minute). Set aside.
For the cake batter: Preheat the oven to 350 degrees F; line a cupcake pan with 12 paper liners.
Sift together the flour, cornstarch, baking powder and salt in a large bowl and set aside. In the bowl of a stand mixer or using a hand mixer, mix the granulated sugar and vegan butter together on medium-low speed until fluffy, 3 to 5 minutes; scrape down the bowl. Slowly mix in the soy milk, applesauce and vanilla, scraping down the bowl a couple of times to make sure everything is fully mixed. Slowly add the flour mixture and mix to combine. Fold in the vinegar and garlic paste by hand, being sure not to over-mix.
Divide the batter among the liners and sprinkle some white chocolate chips on top. Bake until an inserted toothpick comes out clean, about 18 minutes. Cool the cupcakes completely before frosting.
For the honey black garlic brittle: Add the granulated sugar and corn syrup to a microwave-safe bowl and microwave on high for 2 minutes; the mixture should be amber or honey-colored. (Or heat until it reaches 300 degrees F on a candy thermometer.) Add the vegan butter and garlic paste and microwave for another minute and a half. Add the baking soda and vanilla and whisk together. Immediately pour the mixture onto a nonstick pan (or a pan lined with aluminum foil). Freeze for 10 to 15 minutes.
For the cardamom frosting: Using a stand mixer with a whisk attachment, blend the vegan shortening and butter until fully incorporated and mixed well, scraping down the bowl a few times, about 5 minutes. Add 1 1/2 cups of the confectioners' sugar, 1 tablespoon of the soy milk and the cardamom and mix for 2 more minutes. Add the remaining 1 1/2 cups confectioners' sugar and 1 tablespoon soy milk, and whip for another minute. If the mixture is too thick, add a splash more of soy milk. If too thin, add 1/4 cup more sugar.
To assemble, pipe some cardamom frosting on each cupcake. Place a shard of the garlic brittle through each fondant heart and place on the cupcakes.
This recipe was created by a contestant during a cooking competition. It has not been tested for home use.