The best unbiased estimate of a zero-count chance

You have independent counts $X_1, \dots, X_n$ from a Poisson$(\lambda)$ and you want the best unbiased estimate of $\theta = e^{-\lambda}$, the probability that a given minute sees zero events.

Starting from the crude estimator "was the first minute empty," build the minimum-variance unbiased estimator by conditioning on the sufficient total $T = \sum_i X_i$.