For Chocolate Cupcakes: Preheat oven to 350F. In a mixing bowl, combine sugar, flour, cocoa powder, baking powder, baking soda, and salt. In another small bowl, combine eggs, milk, vegetable oil, vanilla, and mix to combine.
With a mixer on low speed, slowly add the wet ingredients to the dry ingredients and mix until combined. Divide the batter between the cupcake pans lined with cupcake papers. Fill 1/3 with cake batter. Bake for 12-15 minutes, cool.
For Buttercream Icing: Place butter, powdered sugar, and cocoa powder in a stand mixer and mix for 3-4 minutes. Start on a low speed for the first minute and then increase to medium speed for the duration. Add cream, salt, vanilla, and mix well.
Assembly: Remove cupcake tops with a knife. Scoop out the center of the cupcake, careful not to remove too much. Fill cavity with 2 tablespoons of Häagen-Dazs® ice cream and replace cupcake top. Freeze.
After about 60 minutes in the freezer, apply the icing to the top of the cupcakes using a knife or pastry bag and refreeze. Keep cupcakes frozen until ready to serve.