  We study the mass of asymptotically flat $3$-manifolds with boundary using
the method of Bray-Kazaras-Khuri-Stern. More precisely, we derive a mass
formula on the union of an asymptotically flat manifold and fill-ins of its
boundary, and give new sufficient conditions guaranteeing the positivity of the
mass. Motivation to such consideration comes from studying the quasi-local mass
of the boundary surface. If the boundary isometrically embeds in the Euclidean
space, we apply the formula to obtain convergence of the Brown-York mass along
large surfaces tending to $\infty$ which include the scaling of any fixed
coordinate-convex surface.
