To a blender add half the milk, half the evaporated milk, the avocado, banana, honey, cocoa and vanilla. Blend until well combined and smooth, about 30 seconds. Add the remaining milk and evaporated milk and blend for 10 seconds until well incorporated.
Drizzle each popsicle mold with 1 teaspoon of chocolate syrup. Divide the milk mixture among the molds, filling each up 1/4-inch from the top. Cover the molds with the lid (insert ice-pop sticks if needed) and freeze overnight. To unmold, dip the mold into warm water for 30 seconds to loosen the ice pops.