How do I dye cookies blue for Channukah without them looking gross. Planning to make star of david channukah cookie sandwiches with frosting to ship

Frosted would probably look the best. But, if you don't want a heavy buttercream, I just made a simple milk and confectioner sugar icing (but on the thicker side), and add the flavoring off your choice. Then you can either color it a solid blue or instead of mixing in the food color, swirl the drops a bit, using a toothpick. This will give a marbling effect on the finished cookie. You can then simply dip the top of your cookie in the icing and place on a baking rack to allow the icing to dry. Very good with a sugar cookie.

This site has some ideas: https://toriavey.com/toris-kitchen/hanukkah-holiday-sugar-cookies/

If you over mix raspberries or blueberries in a dough it turns blue... But, I agree with the other comments. Even if you have blue dough- when it bakes it just looks unappealing and vile.

I also wouldn't dye the dough itself. Blue is a tricky color for food. The star of David shape would be enough to channel Chanukah for me, but if you really want blue, I might go with a line of piping, with a sprinkle sanding sugar, or with gazing/dipping/flooding the tops (or a portion thereof) in blue.

How about decorating with blue and white icing?

I agree with Nancy about not coloring the actual cookies blue.
How about using blue sanding/sparkling sugar on top of the cookies?

Can't think of way to dye them blue and be appetizing.
Maybe use alternate design scheme?
If they are cut-out to be 6-pointed star, just add silver or gold dragees on top of white icing.
If they not star-shaped, ice a star on top of cookie, then add glittering dragees.