Currently there are no immediate solutions (that I can think of) to the 'variable width walls from edge-based mesh' problem, though it is interesting. There are however Offset Scripts that generate faces from edge-based meshes, but they don't work with the T junction, only edge-loops.
A potentially doable approach is to first make your maze shape using an edge based mesh. Then convert it to a Curve (Ctrl+C) (Curve from Mesh)
Then tweak the Bevel Depth to get the preferred thickness (radius). Then stick on a Remesh Modifier (or two - Blocks seems to be a good mode), but you ought to experiment which it until it comes close to what you want. If you want thinner walls you decrease the Bevel depth of the Curve, and adjust the Scale and Octree Depth in the Remesh mmodifier.