N.Y. HAY Law § 185

§ 185. Difference about improvements. When the town superintendent or\nthe officers of a village or city having the powers of town\nsuperintendents therein, shall desire to make a new or altered highway\nextending beyond the bounds of such town, village or city, a better\nhighway than is usually made for a common highway, with a special grade\nor roadbed, drainage or improved plan, and are willing to bear the whole\nor a part of the expense thereof beyond such bounds, but cannot agree in\nregard to the same, upon written application of either of the\nsuperintendents or officers and notice to all parties interested, such\ncourt shall make an equitable adjustment of the matters, and may direct\nthat in consideration of the payment of such portion of the additional\nexpense by the town, village or city that desires the improved and\nbetter highway, as shall be equitable, its officers, contractors,\nservants and agents may go into such town, village or city, and make the\ngrade and roadbed, and do whatever may be necessary and proper for the\ncompletion of such better highway, advancing the money to do it; the\namount of damages to each owner or occupant shall be ascertained and\ndetermined by the supreme court in accordance with the eminent domain\nprocedure law, such court shall, on notice to all parties interested,\ndirect that the amount of damages assessed each owner or occupant, if\nany and all such expenses be paid by each, any or all of such towns,\nvillages or cities as shall be just and equitable, and the damages and\nexpenses assessed and allowed, as in this and the last preceding\nsections, shall be paid and collected as if fixed by the town\nsuperintendents of the towns, or the officers of such villages or cities\nhaving the powers of such superintendents.\n