I asked this question on M.SE a while ago and got no answers, so I'm asking it here.

Let $\kappa$ be an infinite cardinal. A family $\mathcal{A}\subseteq\mathcal{P}(\kappa)$ is independent if for any $A_1,\ldots,A_n\in\mathcal{A}$ and $i_1,\ldots,i_n\in\{0,1\}$, we have $$ \left|\bigcap_{k=1}^n A_k^{i_k}\right| = \kappa $$ where $A^0 = A$ and $A^1 = \kappa\setminus A$.

Question: Is there an independent family $\mathcal{A}$ such that the Boolean algebra generated by $\mathcal{A}$, along with the subsets of $\kappa$ of size $< \kappa$, is all of $\mathcal{P}(\kappa)$?

I am particularly interested in the case $\kappa = \omega_1$, though an answer for any $\kappa$ would be interesting.