I have found that it is very important to start mowing what I have cleared. Several time I cleared an area, didn’t mow it for a year or two and had to clear it again (yeah sometimes I learn slow). Close to 10 years ago I happened to find a used DR Field & Brush mower for sale. While it takes a while to mow an acre of just grass, it does great in light brush and will even take down a sapling. But such a mower will not take down a stand of brush, at least not without a whole lot of pushing and taking it a bit at a time.
This picture shows an area that I cleared a path through and then kept mowing it every month or so through the growing season. You can see that some kind of grass will start growing and it makes a very pleasant place to walk through.