  We construct smooth solutions to Ricci flow starting from a class of singular
metrics and give asymptotics for the forward evolution. The singular metrics
heal with a set of points (of codimension at least three) coming out of the
singular point. We conjecture that these metrics arise as final-time limits of
Ricci flow encountering a Type-I singularity modeled on $\mathbb{R}^{p+1}
\times S^q$. This gives a picture of Ricci flow through a singularity, in which
a neighborhood of the manifold changes topology from $D^{p+1} \times S^{q}$ to
$S^p \times D^{q+1}$ (through the cone over $S^p \times S^q$.)
  We work in the class of doubly-warped product metrics. We also briefly
discuss some possible smooth and non-smooth forward evolutions from other
singular initial data.
