Dough: Sprinkle yeast over warm water. Add 1 teaspoon of sugar and stir to combine. Let sit for 5 minutes to allow mixture to bubble and activate.
In the bowl of a stand mixer, whisk together yeast mixture, remaining sugar, egg, and egg yolk. Attach dough hook to mixer.
In a large bowl whisk together flour and salt. Add 2 cups of flour to mixing bowl. Mix on low speed until dough starts to come together. Add another 1 cup of flour. Continue to knead on low speed and add softened butter one tablespoon at a time. Add remaining cup of flour and knead until dough is smooth and cleans the sides of the bowl. Add chopped Peppermint Bark and mix until evenly distributed in dough.
Transfer dough to a lightly greased bowl. Cover with plastic wrap and let dough rest at room temperature for one hour until dough has doubled in volume.