  Let $M$ be a compact 3--manifold with boundary a single torus. We present
upper and lower complexity bounds for closed 3--manifolds obtained as even Dehn
fillings of $M.$ As an application, we characterise some infinite families of
even Dehn fillings of $M$ for which our method determines the complexity of its
members up to an additive constant. The constant only depends on the size of a
chosen triangulation of $M$, and the isotopy class of its boundary.
  We then show that, given a triangulation $\mathcal T$ of $M$ with
$2$--triangle torus boundary, there exist infinite families of even Dehn
fillings of $M$ for which we can determine the complexity of the filled
manifolds with a gap between upper and lower bound of at most $13 |\mathcal T|
+ 7.$ This result is bootstrapped to obtain the gap as a function of the size
of an ideal triangulation of the interior of $M$, or the number of crossings of
a knot diagram. We also show how to compute the gap for explicit families of
fillings of knot complements in the three-sphere. The practicability of our
approach is demonstrated by determining the complexity up to a gap of at most
10 for several infinite families of even fillings of the figure eight knot, the
pretzel knot $P(-2,3,7)$, and the trefoil.
