I can think of two ways

1. Dig out the foot prints add 3 inches of soil and then replace the pieces you've dug out .That might give you a fairly quick fix but might be difficult to get entirely level unless done very carefully

2. spread some fine soil into the holes to fill them up and reseed . Grass seed normally takes about 9 days to germinate if kept watered and the holes should be fairly well covered with new grass within a month

I would use Anchorman's option 2.

So would I!

Yes, I would say 2 too.
All this takes me back to when I was a little girl and lived with dairy farms all around us ( Alex Steels) farm was one, we lived in the country a place called Shopland Nr Rochford Essex, loved it there, school was at the bottom of the garden, only a 100 children went there, played with the farmers children, drank milk from the coolers that had just come from the cows, picked mushrooms in the fields, such a happy childhood my brother & I had, I remember opening the kitchen door & there was a few cows in our garden, they had broken down the fence in the field & yes, their hoof prints were very deep.
Oooo! Sorry for going on but this posting brought back so many childhood memories :o))

Hi Kendo, whilst agreeing with the others about the way to repair the damage, I would question whether the ground was prepared properly in the first place.
To prepare the ground for a lawn, it should first be dug and weeded, raked and levelled, firmed by going all over it using your heels to firm as much as possible, raked again, and even again ,until you're certain that you've made it as even as possible, and all large stones etc have been removed, when this is completed the soil should not go down 3 inches whatever stands on it, Derek.

Generally I agree but it depends on how wet the ground is. Some horses got into an ancient lawn of mine which had had decades to settle and left 3 inch deep hoof marks