  We consider Burgers equation with transverse viscosity
$$\partial_tu+u\partial_xu-\partial_{yy}u=0, \ \ (x,y)\in \mathbb R^2, \ \
u:[0,T)\times \mathbb R^2\rightarrow \mathbb R.$$ We construct and describe
precisely a family of solutions which become singular in finite time by having
their gradient becoming unbounded. To leading order, the solution is given by a
backward self-similar solution of Burgers equation along the $x$ variable,
whose scaling parameters evolve according to parabolic equations along the $y$
variable, one of them being the quadratic semi-linear heat equation. We develop
a new framework adapted to this mixed hyperbolic/parabolic blow-up problem,
revisit the construction of flat blow-up profiles for the semi-linear heat
equation, and the self-similarity in the shocks of Burgers equation.
