One-handle cobordism from a disjoint union to a connected sum

Start with $(X_0\bigsqcup X_1)\times [0,1]$ and attach an $(n+1)$-dimensional one-handle $D^1\times D^n$ to one $n$-ball in each component of its outgoing boundary. The new outgoing boundary is $X_0\#X_1$, so this is a cobordism from $X_0\bigsqcup X_1$ to their connected sum.