Preheat oven to 350°F (180°C). Line a 12-serving cupcake pan with paper liners; set aside.

In a large mixing bowl, beat 3/4 cup granulated sugar and 2 large eggs with an electric mixer at medium speed for 1 minute. Add 1 1/4 cups gluten-free flour, 1/4 teaspoon salt, 1 3/4 teaspoons gluten-free baking powder, 1/2 cup oil, 1/2 cup dairy-free milk, and 2 teaspoons peppermint extract. Continue to beat at medium speed for 1 more minute.

Divide batter among liners, filling 2/3 full (about 1/4 cup of batter per cupcake). Bake for 23-25 minutes until a toothpick comes out clean. Cool completely before frosting.

For the frosting: In a medium mixing bowl, beat 1/4 cup dairy-free butter and 1/4 cup palm shortening with an electric mixer until creamy. Add 1 teaspoon peppermint extract and mix until combined. Beat in 1 cup of powdered sugar at a time, alternating with 1 tablespoon of dairy-free milk. Add more or less dairy-free milk for desired consistency. Pipe frosting onto each cupcake. Sprinkle with crushed candy cane pieces.