A petition for an order for the laying out, repair, resurfacing or other improvement of any highway, which is the dividing line between two towns or adjoins the dividing line between two towns, or is located wholly within one town and land fronting thereon is located in another town, may be brought to a judge of the Superior Court by either of such towns if, in the opinion of the official or officials having charge of the repair or maintenance of highways in either of such towns, any such highway should be laid out, resurfaced or otherwise improved, and such towns are unable to agree as to whether such work should be done or as to the method of payment therefor. Such judge, after reasonable notice to all parties interested, if he finds that such highway should be laid out or such improvement should be made, shall order the layout of such highway or the making of such improvement, as the case may be, and provide in such order which of such towns shall make such layout or improvement and shall fix the proportion of the expense thereof which shall be paid by each of such towns.