This talk concerns a relationship between two much-studied classes of models  of motion in a random medium, namely  random walk in  random environment (RWRE) and the Kardar-Parisi-Zhang (KPZ)  universality class.  Barraquand and Corwin discovered that in 1+1 dimensional RWRE in a dynamical beta environment the correction to the quenched  large deviation principle  obeys KPZ behavior.   In this talk we condition the  beta walk to escape at an atypical velocity and show that the resulting Doob-transformed RWRE obeys the KPZ wandering exponent 2/3.   The logarithm of the harmonic function in the Doob transform obeys the KPZ 1/3 exponent.   Based on joint work with M\'arton Bal\'azs (Bristol) and Firas Rassoul-Agha (Utah).