We investigate the regularizing effect of adding small fractional Laplacian, with critical fractional exponent 1/2 , to a general first
 order HJB equation. Our results include some regularity estimates for the viscosity solutions of such perturbations, making the solutions 
 classically well-defined. Most importantly, we use these regularity estimates to study the vanishing viscosity approximation to first order 
 HJB equations by 1/2-Laplacian and derive an explicit rate of convergence for the vanishing viscosity limit.