Can I still use apple cider vinegar in cooking recipes if there is a giant disk of vinegar mother at the top?

The mother of the vinegar is essentially a curd of dead bacteria and cellulose. Once the vinegar has become vinegar and the curd has appeared. Gently pour the vinegar through a coffee filter! That will catch any of the curd in the coffee filter.

Now rinse the curd out of the bottle. And pour your vinegar back in to the bottle. The curd is not poisonous it's just fermented bacteria protein and some cellulose.

And they call that the mother of the vinegar because it was the mother of the vinegar it was the bacteria that converted the alcohol and the sugar into the acetic acid and then it died and now it's a curd

Absolutely,  .... skim. or strain it out, and use it......  the  " Vinegar Barrell"  Has "mother"  floating  at the top,  It's  part of Nature....Store bought Vinegar has been strained and filtered and had the Acidity, (strength) adjusted

Just strain out the mother disc. You don't get that from commercial vinegar. 

According to google, which knows everything, it apparently is safe to consume the vinegar.

You can use the vinegar, just strain out the Mother

Maybe a small-screen strainer would help. 

Either use it or pay almost $5 a quart for more cider vinegar-  gallon is still cheap but huge shortage of plain old cider vinegar in qts.