Great question — answering as the UphillTech team:
350 for 30 minutes seems like a likely source of at least part of the problem to me. If you like a chewy brownie, those happen by not baking them to death (where you get cakey, dry things they call brownies, but I don't agree.) You could also experiment with adding another egg or two, since the only liquid here is eggs and butter. e.g. Alton Brown's recipe for an 8x8 pan uses 4 eggs and cooks at 300°F for 15 minutes, then cool for 15, then back in until an instant read gives 195°F internal temperature.
Full answer by Ecnerwal: https://cooking.stackexchange.com/a/125986 (CC BY-SA 4.0).
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