I've never grown carrots, but I know they need loose soil so they can push roots down into it.
Is the soil under your brush pretty soft already? If it's hard clay or rocky, killing the brush and adding compost might be only the first step. You might need to remove rocks so they can grow straight, or amend hard clay so they can grow longer than an inch or two.
I've planted long radishes (Daikon radishes) in hard clay, and they do manage to push down a ways, but they also stick up out of the ground as if the clay was pushing back. I don;lt thnik carrots are that aggressive.
If the soil does need to be amended for carrots, but you aren't eager to get out a pick, shovel and garden fork, you might plant some clay-tolerant cover crop the first year to do some of the initial "tilling" and breaking up of hard clay. "Tillage radishes" or anything with aggressive roots might help.
