Line a large baking sheet with parchment paper. Skewer each banana half with a lollipop stick and freeze until firm, at least 2 hours.
In a large microwave-safe measuring cup, melt 1 pound white chocolate in the microwave according to package directions. One at a time, dip the bananas into the chocolate to coat, shaking off any excess. Return to the baking sheet and immediately place two M&M candies near the top for eyes.
Melt the remaining white chocolate in the microwave according to package directions and fill a small piping bag fitted with a small, round nozzle tip. Pipe zigzags over the banana to form bandages. Repeat this step with the milk chocolate, if using. Chill or freeze until ready to serve.