In the past years, I've tried to achieve a basketweave-look on cookies using a multitude of techniques.
I think I've finally found a way that I'm very pleased with.

Using this awesome basketweave embosser, you can create the
effect directly on the cookie dough or over partially dried glaze.

Here's an example of embossing cookie dough before baking
and then making a cookie dough braid with this mold and adding it to the top of the cookie.

Once the cookie is cool after baking, brush it with various brown dusts to give it a rich basket look.
(Colors used here: caramel, deep brown, and chocolate)

To emboss partially dried glaze, outline and flood the cookie with glaze.
Allow it to dry for 4-5 hours.
Firmly press the embosser into the glaze.

Dry overnight.

Brush the weave with various brown dusts as described above.
(Colors used here: caramel, deep brown, and chocolate for the brown basket,
and less dust for the white basket.)

Mold some colorful "eggs" or flowers with edible clay to add to the top of the basket cookie.