I have done a lot of lawns like this and there are several options. The best way I would go about it is to rototill the area, rake or grade smooth and seed. The idea is the existing grass root system is difficult to deal with and tilling it knocks down the high spots and fills the low spots. As i said, this is ideal and my first choice.
Next in line is to fill the low spots with dirt by hand and rake smooth, but that's a lot of work and will still leave an irregular lawn. A better choice for the dirt idea is to spread the dirt by hand and use a landscape (rock) rake with the tractor to drag around the dirt to fill the low spots and, to a degree scrape off the high spots. The vibrating times tear off some of the high spots and spread compacted dirt around better. A little seed and this works pretty well. I have never used a boxblade with this approach but I think it would work well by spreading dirt around and pulling over the surface, not pushing. Tines/teeth would not be used.
If you spread dirt around it needs to be very dry or it won't spread well and will hump up into little piles that are hard to deal with. Same thing if you dump it with a bucket loader. It will work if the dirt is very dry and you "shake" it out of the bucket a little at a time, move the humps around by hand and immediately grade it.