If you consider the curves as a graph, a path which travels along each edge exactly once is known as an Eulerian path.
Such a path only exists in a graph where a maximum of 2 of the vertices have an odd degree (the number of connected edges).
If you think of one continuous path, apart from the vertex it starts and ends at, to visit a vertex without travelling along any edge more than once, it needs one edge to enter the vertex and one edge to leave it. It can visit the vertex multiple times, but each time it needs to use 2 of the connected edges, hence the degree must be even.
Unfortunately the graph in your image above has nearly all degree 3 vertices, so it isn’t possible to mill in a continuous path without going over some edges more than once.