  This paper is devoted to studying the Hamilton-Jacobi-Bellman equations with
distribution-valued coefficients, which is not well-defined in the classical
sense and shall be understood by using paracontrolled distribution method
introduced in \cite{GIP15}. By a new characterization of weighted H\"older
space and Zvonkin's transformation we prove some new a priori estimates, and
therefore, establish the global well-posedness for singular HJB equations. As
an application, the global well-posedness for KPZ equations on the real line in
polynomial weighted H\"older spaces is obtained without using Cole-Hopf's
transformation. In particular, we solve the conjecture posed in \cite[Remark
1.1]{PR18}.
