Line a cookie sheet with parchment paper, set aside.
Place about ¾ of the white chocolate chips in a microwave-safe bowl. Add the vegetable oil. Melt the white chocolate chips in the microwave in 30-second increments. Stir the chocolate with a spatula or wooden spoon between each blast. It should be melted and glossy.
Spoon the melted white chocolate over the Nutter Butter cookies or dip one side in the chocolate. Cover one side of each cookie completely. Repeat with the remaining cookies.
Place on parchment paper and allow to harden completely.
Add the candy eyes while the chocolate is still wet. Allow the white chocolate to harden.
Melt the remaining chocolate. Spoon it into a piping bag. Snip off the tip. Drizzle the melted chocolate back and forth over the cookies so they look like little mummies.
Allow the chocolate to harden again before stacking.
Store in an airtight container at room temperature until ready to serve.
** Do not place them in the refrigerator; the candy eyes will bleed!