I've never tried it, but would think that the glue might gum up the Cricut, and that would be bad. You could cut the letters out of the chipboard, then the paper, and then adhere them, though...would be slightly more work, but wouldn't risk wrecking the machine.
You'd need to cut the cardstock and chipboard separately as Bree said, but you need two different blades...one for the cardstock (green houseing) and one for the chipboard (blue housing)...the regular green one will not cut chipboard.
That's what I'll do -- cut the paper and the chipboard separately (with different blades) and then run the paper letters through the Xyron and adher them to the cardstock.
you don't have to have the deep cut blade to cut chipboard. i use my regular one but you DO need it if you are cutting magnetic paper/stickers or thicker type chipboard.