If it is coming from under the hood, I’d guess the brake booster has sprung a leak. If your car even has brake shoes, they’d be on the rear wheels, not in the front. I can’t see this problem being caused by brake shoes from your description.
This is a safety issue and needs to be addressed. Nothing should be making a gurgling noise when you press on the brakes. This isn’t a good problem for using a dealership. Suggest you ask friends, relatives, neighbors, co-workers etc for a recommendation for an independent mechanic who specializes in Japanese or Import cars. The Corolla is probably the most popular car in America, so it should be easy to find a good shop. Best of luck.