  We develop a martingale approach for a class of singular stochastic PDEs of
Burgers type (including fractional and multi-component Burgers equations) by
constructing a domain for their infinitesimal generators. It was known that the
domain must have trivial intersection with the usual cylinder test functions,
and to overcome this difficulty we import some ideas from paracontrolled
distributions to an infinite dimensional setting in order to construct a domain
of controlled functions. Using the new domain, we are able to prove existence
and uniqueness for the Kolmogorov backward equation and the martingale problem.
We also extend the uniqueness result for "energy solutions" of the stochastic
Burgers equation of [GP18a] to a wider class of equations.
