Preheat oven to 350 ° F.
In a stand mixer fitted with a paddle attachment, cream butter and sugar until smooth. Add the egg and beat until completely incorporated. While the mixer is running on low, add in flour and mix until just combined.
Shape dough into 1-inch balls (about 2 tablespoons each).
Place on parchment-lined baking sheets, at least 3 inches apart. If the dough is too warm to form into smooth balls, place the dough in the fridge for a few minutes to set up again.
Moisten thumb with water and gently press the center of each ball, making an indentation about 1/2 inch wide and an inch deep.
In a small pot, heat jam until liquefied; spoon about 1/2 teaspoon into each indentation. Sprinkle the outside edge of the cookie with the crushed peppermint candy, lightly pressing it in to make sure it stays affixed. Bake until cookies are golden brown around edges, 18 to 20 minutes. Transfer to a wire rack and let cool completely.