Here [1] is a CLT for RV with infinite variance, Prop 3.1.12, but notice the (larger) scaling coefficient (1/sqrt(n log n)).
Also see the second answer on SO here [2].
[1] https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes....
[2] https://stats.stackexchange.com/questions/169611/the-role-of...
EDIT to add: Having said that, the Lindeberg-Feller and the Lyuapunov formulation of the CLT do require finite variance, so maybe I was too quick in stating that that assumption can be relaxed.