  Consider a Kleinian singularity $ \mathbb{C}^2/\Gamma $, where $ \Gamma $ is
a finite subgroup of $ SL_2(\mathbb{C}) $. In this paper, we introduce a
natural stack compactifying the singularity by adding a smooth stacky divisor,
and we show that sets of framed sheaves on this stack satisfying certain
additional criteria are closely related to a class of Nakajima quiver
varieties. This partially extends our previous work on punctual Hilbert schemes
of Kleinian singularities.
