In this paper we consider a probability distribution on plane partitions, which arises as a one-parameter generalization of the q^{volume} measure. This generalization is closely related to the classical multivariate Hall-Littlewood polynomials, and it was first introduced by Vuletic. We prove that as the plane partitions become large (q goes to 1, while the Hall-Littlewood parameter t is fixed), the scaled bottom slice of the random plane partition converges to a deterministic limit shape, and that one-point fluctuations around the limit shape are asymptotically given by the GUE Tracy-Widom distribution. On the other hand, if t simultaneously converges to its own critical value of 1, the fluctuations instead converge to the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with the so-called narrow wedge initial data. The algebraic part of our arguments is closely related to the formalism of Macdonald processes. The analytic part consists of detailed asymptotic analysis of the arising Fredholm determinants.