It is not usually considered practical to attempt to eradicate mushrooms growing in your lawn. Sometimes the normal application of lawncare products (fertilizer and weedkiller) may kill them off anyway.

Are there any dead tree roots underlying your lawn (eg from a tree cut down a few years ago)? I have some fairly large fungi in my lawn feeding on large rotting underground roots, and I reckon that once they have done their job in breaking down the wood, they'll be gone. They look a bit brain-like.

There are many wild mushrooms that don't look like the ones on your pizza that are highly desirable to eat, even some that look like brains. Morels (Morchella) are often described as brain-like, highly desirable, but they are too small for your description and it is the wrong time of year for them. More plausibly, it could be Sparassis crispa (cauliflower mushroom), which is also very desirable, provided you pick it early enough, otherwise it goes rubbery. There are also plenty of other things it could be. So make sure you get an absolutely positive identification before eating it as a few wild mushrooms are deadly poisonous, even if cooked.