In 2017, Finkelberg and Tsymbaliuk introduced the notion of shifted quantum affine algebras and described their role in the study of K-theoretic quantized Coulomb branches associated with certain 3D $N=4$ quiver gauge theories. We describe a new geometrical construction of a deformation of one of these shifted quantum affine algebras as the Hall algebra of the category of restricted representations of the Lie algebra $\mathfrak{sl}_2$ over a finite field. The main tool we use is an equivalence of categories by Rudakov that relates the above category to that of representations over a certain quiver modulo relations. This is joint work with Peter Samuelson.