I did do a tiny bit of research after the fact. I do believe that I:
1. didn't beat my egg whites long enough,
2. didn't bake the shells long enough, and
3. have an oven that heats unevenly.
Needless to say, because the shells didn't work out, I didn't bother making the cheese cream buttercream filling. So we've been eating a lot of bagels lately...

I might just stick to buying them...