We too brood in the covered run. Deep bedding, no cleaning.I brood out in my run, directly on the deep litter, with a sprinkle of aspen shavings on top just to make for a softer sleeping space under the heating pad.
I know this set up isn't for everyone, but I can't see setting up a brooder any other way. Nearly no cost (a few handfuls out of a bag of shavings is a minimal expense) and zero clean up. Even setting up is pretty easy: kick loose litter into a big pile, dragging the brooder on top of it, and that's about it.
Here's a decent look inside it, right after I "cleaned" by dragging it across the run.
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We are in SE Mass. At 10 days, they spend very little time under the MHP.