  We study a $b$-deformation of monotone Hurwitz numbers, obtained by deforming
Schur functions into Jack symmetric functions. We give an evolution equation
for this model and derive from it Virasoro constraints, thereby proving a
conjecture of F\'eray on Jack characters. A combinatorial model of non-oriented
monotone Hurwitz maps which generalizes monotone transposition factorizations
is provided.
  In the case $b=1$ we obtain an explicit Schur expansion of the model and show
that it obeys the BKP integrable hierarchy. This Schur expansion also proves a
conjecture of Oliveira--Novaes relating zonal polynomials with irreducible
representations of $O(N)$. We also relate the model to an $O(N)$ version of the
Br\'ezin--Gross--Witten integral, which we solve explicitly in terms of
Pfaffians in the case of even multiplicities.
