For marbled cookies, you need Royal Icing. Royal Icing is a fairly runny icing that dries smooth and hard. For swirls, divide your Royal Icing in two bowls. Tint one, and leave the other white (or tint it a contrasting color). Frost cookies with a solid-color base. The first technique is dipping the cookie facedown into the bowl of icing. Let the excess drip off, then quickly flip the cookie over and set it on a rack.
A third technique for icing a base coat is to pipe a border around the cookie using a piping bag or zip-top bag with the tip of one corner snipped off. You can do the border in a contrasting color (pictured) or in the same color as the base color.
For a spider web design, pipe a contrasting color over the base color. (The base color should still be wet.) Pipe a spiral starting in the center or concentric circles (pictured). You can also experiment with stripes or polka dots. Each will create a different marbled pattern.
Starting in the center, gently pull a toothpick through both colors of icing to swirl the colors together. If you’ve piped polka dots, pull the toothpick in random curves for a casual marbled design. If you’ve piped stripes, pull the toothpick in straight lines perpendicular to the piped stripes for a scalloped pattern. Wipe the toothpick clean as needed.