What should I do if I found a really big tick that’s burrowed into my baby’s skin?

Remove it gently. First of all...Don't panic! Ticks are common, and most bites are harmless, but you do want to remove them because they can spread disease. The easiest way to remove the tick is to take tweezers, grab the tick at the head or mouth (closest to the skin), and apply gentle pressure upwards (away from skin). Do not use a match or petrolatum, since this will cause the tick to burrow deeper.
Remove it. If you are not squeamish grab the tick with a tweezers pulling straight out with constant tension. Do not twist. If you have frequent encounters with ticks there are devices that you can slide under the tick and yank it out, as you would a nail with the back side of a hammer. Then clean the area with alcohol.