  We consider the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nabla
u|^p,$$ with Dirichlet boundary conditions in two space dimensions, which
arises in the KPZ model of growing interfaces. For $p>2$, solutions may develop
gradient singularities on the boundary in finite time, and examples of
single-point gradient blowup on the boundary are known, but the space-profile
in the tangential direction has remained a completely open problem. In the
parameter range $2<p\le 3$, for the case of a flat boundary and an isolated
singularity at the origin, we give an answer to this question, obtaining the
precise final asymptotic profile, under the form $$u_y(x,y,T) \sim
d_p\Bigl[y+C|x|^{2(p-1)/(p-2)}\Bigr]^{-1/(p-1)},\quad\hbox{as $(x,y)\to
(0,0)$.}$$ Interestingly, this result displays a new phenomenon of strong
anisotropy of the profile, quite different to what is observed in other blowup
problems for nonlinear parabolic equations, with the exponents $1/(p-1)$ in the
normal direction $y$ and $2/(p-2)$ in the tangential direction $x$.
Furthermore, the tangential profile violates the (self-similar) scale
invariance of the equation, whereas the normal profile remains self-similar.
