Preheat oven to 350°F. Line two baking sheets with parchment paper.
Whisk flour, cocoa, baking powder, and salt in a bowl; set aside.
Beat softened butter and sugar until light and fluffy, 2 to 3 minutes.
Mix in vanilla and milk until smooth.
Add dry ingredients and blend on low until a soft dough forms; do not overmix.
Scoop 1-tablespoon portions, roll into balls, and coat in coarse sugar if using.
Arrange on sheets, spacing 2 inches apart. Press a deep well in each with your thumb or a 1/2-teaspoon measure.
Fill each indentation with a small spoonful of cherry preserves, avoiding overflow.
Bake until edges are set and tops look dry, 10 to 12 minutes.
Cool on the sheet for 5 minutes, then move cookies to a rack to cool completely.
Melt chocolate chips with oil until smooth; drizzle over cooled cookies and let set.