Do I need to get permission from the King himself? Local Lords? Do I provide the labor or will they at my expense? Can I charge a toll? Am I even allowed to build infrastructure as a commoner?
While more can always be said on the topic, my following posts are in fact to offer some backgrounds to consider the problem:
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In short, if you pay a toll to some lords by the ravine and the road is generally "public", it should be their obligation to maintain less troublesome traffics across the ravine. You can ask them for better roads, including the bridge and pavement.
While the emperor or the king is generally responsible in theory for the ultimate source of authority of maintaining the public roads after the later 12th century, they sometimes delegated this obligation [together with the privilege of collecting the toll] to local lords, but as long as you have to pay the toll at the post, it would not be difficult to identify who to ask.
If the lord is indifferent to your request, you should perhaps consider the possibility of switching to alternative route where other lords were more eager to offer their "service" on roads in exchange for your toll......