N.Y. High. Law § 185
When the town superintendent or the officers of a village or city having the powers of town superintendents therein, shall desire to make a new or altered highway extending beyond the bounds of such town, village or city, a better highway than is usually made for a common highway, with a special grade or roadbed, drainage or improved plan, and are willing to bear the whole or a part of the expense thereof beyond such bounds, but cannot agree in regard to the same, upon written application of either of the superintendents or officers and notice to all parties interested, such court shall make an equitable adjustment of the matters, and may direct that in consideration of the payment of such portion of the additional expense by the town, village or city that desires the improved and better highway, as shall be equitable, its officers, contractors, servants and agents may go into such town, village or city, and make the grade and roadbed, and do whatever may be necessary and proper for the completion of such better highway, advancing the money to do it; the amount of damages to each owner or occupant shall be ascertained and determined by the supreme court in accordance with the eminent domain procedure law, such court shall, on notice to all parties interested, direct that the amount of damages assessed each owner or occupant, if any and all such expenses be paid by each, any or all of such towns, villages or cities as shall be just and equitable, and the damages and expenses assessed and allowed, as in this and the last preceding sections, shall be paid and collected as if fixed by the town superintendents of the towns, or the officers of such villages or cities having the powers of such superintendents.
N.Y. High. Law § 185