How do I apply compost on a large lawn?

I would load it into a wheelbarrow and use an empty coffee can to have greater control spreading it.

Once spread, it should be tilled in to a depth of several inches. This is what we did. The spreading doesn't need to be particularly even as the tiller mixes it. I think we just dumped wheel barrow loads and moved it around a little with a landscape rake. Then tilled after it was all down.

I agree with Lucky Home by hand-- if you do it this way no raking to level or getting it too heavy in some areas which will kill/suffocate grass

You can use a lawn drop spreader or a broadcast spreader. The drop spreader drops the compost straight down, whereas the broadcast spreader slings the compost to the front and sides of the spreader. Both are adjustable so you can spread it as thick or thin as you like.

https://www.amazon.com/s?k=Lawn+seeder&ref=nb_sb_noss

Have you ever made compost tea? It's a lot easier to sprinkle it on your lawn and does not require wheel barrows, or shovels.