Preheat the oven to 350 degrees. Line two baking sheets with parchment paper and then set aside.
Blend together ½ cup of softened butter and the sugar. Cream together until light in color and fluffy. This can be done with a stand mixer or a hand mixer.
Add in the vanilla and egg. Mix together well
Add in the flour and stir together until the mixture forms a dough and pulls away from the side of the bowl. If you are using a stand mixer, change the whisk attachment to the paddle attachment.
Divide the dough in half.
Combine one half of the cookie dough, the cocoa powder and the remaining ¼ cup softened butter. Mix together until smooth. Set aside.
With the remaining half of the cookie dough, divide the portion into three smaller, equal sections. Squeeze a couple drops of food coloring (yellow, pink and blue) into each of the three sections, with one color added to each section.
Gently blend the food coloring into the color by kneading just until the dough is streaked with color. You do not want to mix in the color completely, but rather a tie-dyed look.
Once each ball of dough has color, roll it into logs.
Place the logs side by side and press together. Reshape the logs if needed to equal out the lengths and width.
Form the chocolate dough into a log, as well.
Now lay the chocolate log on top of the colored log and gently press together until they are combined and do not fall apart, but are still separate colored logs.
Using a sharp knife, cut off about ¼ inch slices from the cookie dough log.
Reshape the cut dough slices into more of a circle shape if needed and place on the baking sheet. Leave a little room around each cookie, but they should not spread much while baking.
Continue with the remaining dough.
Bake in the preheated oven for no more than 12 minutes. Do not overbake! It will dry out the cookies. Cookies are done when the edges are set and the edges and bottom are just slightly starting to brown.
Transfer to wire racks and allow to cool completely.
Once the cookies have cooled, add the chocolate chips to a microwave safe bowl. Heat for 30 seconds and stir thoroughly. Heat for another 15 seconds and stir again. Continue the process of heating in short intervals and stirring afterwards so that the chocolate does not burn.
Once melted, transfer to a piping bag or a plastic zip-top baggie. Cut a small hole in the end of the piping bag or plastic bag corner.
Pipe a large circle around the edge of each cookie. Then divide the cookie in half and pipe a line of chocolate between the chocolate side and the colored side. Pipe two more lines equally spaced to create 6 pie slices. Then pipe rounded pieces to connect each line in the form of a spider web.
Allow the chocolate to cool and set.