The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. The double-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours, each of which approximates the optimal solution within at most a factor of 2. We consider the problem of finding among these tours the one that gives the closest approximation, i.e.\ the \emph{minimum-weight double-tree shortcutting}. Burkard et al. gave an algorithm for this problem, running in time $O(n^3+2^d n^2)$ and memory $O(2^d n^2)$, where $d$ is the maximum node degree in the rooted minimum spanning tree. We give an improved algorithm for the case of small $d$ (including planar Euclidean TSP, where $d \leq 4$), running in time $O(4^d n^2)$ and memory $O(4^d n)$. This improvement allows one to solve the problem on much larger instances than previously attempted. Our computational experiments suggest that in terms of the time-quality tradeoff, the minimum-weight double-tree shortcutting method provides one of the best known tour-constructing heuristics.