Summary: In this paper, we consider two models in the Kardar-Parisi-Zhang (KPZ) universality class, the asymmetric simple exclusion process (ASEP) and the stochastic six-vertex model. We introduce a new class of initial data (which we call shape generalized step Bernoulli initial data) for both of these models that generalizes the step Bernoulli initial data studied in a number of recent works on the ASEP. Under this class of initial data, we analyze the current fluctuations of both the ASEP and stochastic six-vertex model and establish the existence of a phase transition along a characteristic line, across which the fluctuation exponent changes from \(1/2\) to \(1/3\). On the characteristic line, the current fluctuations converge to the general (rank \(k)\)) Baik-Ben-Arous-Péché distribution for the law of the largest eigenvalue of a critically spiked covariance matrix. For \(k=1\), this was established for the ASEP by Tracy and Widom; for \(k>1\) (and also \(k=1\), for the stochastic six-vertex model), the appearance of these distributions in both models is new.