  We show how Event Horizon Telescope (EHT) observations of the supermassive
object at the center of M87 can constrain deviations from General Relativity
(GR) in a relatively model-independent way. We focus on the class of theories
whose deviations from GR modify black holes into alternative compact objects
whose properties approach those of an ordinary black hole sufficiently far from
the would-be event horizon. We examine this class for two reasons: ($i$) they
tend to reproduce black-hole expectations for astrophysical accretion disks
(and so do not undermine the evidence linking black holes to active galactic
nuclei); ($ii$) they lend themselves to a robust effective-field-theory
treatment that expands in powers of $\ell/r$, where $\ell$ is the fundamental
length scale that sets the distance over which deviations from GR are
significant and $r$ is a measure of distance from the would-be horizon. At
leading order the observational impact of these types of theories arise as
modifications to the transmission and reflection coefficients of modes as they
approach the horizon. We show how EHT observations can constrain this
reflection coefficient, assuming only that the deviations from GR are small
enough to be treated perturbatively. Our preliminary analysis indicates that
such reflection coefficients can already be constrained to be less than of
order 10\% (corresponding to $\ell \lesssim 100 \mu m$), and so can rule out
some benchmark cases used when seeking black-hole echoes. The precise bounds
depend on the black hole spin, as well as on detailed properties of the
reflection coefficient (such as its dependence on angular direction).
