@Igor Yes! Section 2 does exactly what I asked for. Also, through the References, I found jstor.org.sci-hub.org/stable/2118620 (Section 3) with a different solution. Both are quite complicated, though! I wish there was a simpler example. I'll work my way through these papers, anyway. Thanks.

@Igor By "standard Euclidean metric outside" I mean it IS the standard metric in each point outside N, not just up to isometry. Your construction is of little use here, because it won't generally produce standard flat space outside of a compact set, not even up to isometry, unless $\int f dA=0$, which implies $f$ has positive AND negative values (or is constantly zero). This can be seen by Gauss-Bonnet, as follows. Suppose the space to be standard $R^{2}$ outside N. Take a polygon enclosing N: its angular excess is 0, so $\int k dA=0$ inside the polygon, where $k$ is the curvature.