Professional bakers call this technique "flooding the cookies." Actually, they use an icing with a slightly thicker consistency for the border, then a thinner icing for the middle of the cookie, and they use professional pastry bags with different tips for the border and center of the cookie.
I find that if you use my sugar cookie icing recipe, you don't need to mess with different icings or special equipment. This sugar cookie icing recipe has a touch of corn syrup in it, which keeps it from spreading too much, yet allows it to retain a nice, smooth consistency that dries easily. And as you can see from the photos, we're not going for cooking school perfection here. This is a fun method for decorating Christmas cookies with kids.