Recipes can be tricky to document, because the earliest extant recipe does not necessarily reflect the popularity or origin of a food. Average people did not write recipes down in books to preserve. Cooking was more of an oral/practice tradition. What we do have are records of what was cooked at specific courts, for specific nobility, trade records, monastic records, and "medical" texts with food and spice recommendations for health, like Hildegard von Bingen's [Physica] (https://books.google.is/books?id=wl6w2cfCKTgC).
I can't tell you about the modern, New York cheesecake as we know it today, but there were several examples of cakes or pies made from fresh cheese mixed with eggs and sugar and baked. The Forme of Cury (compiled by the "Master-Cooks of King Richard II") offers a recipe for Sambocade, a pie in which you make a crust (this was generally done with dried bread crumbs and butter), then filling it with a mixture of fresh cheese curds, sugar, and egg whites. It was flavored with dried elderflowers and rose water. I find it a bit more crumbly than modern cheesecake, but similar.
I've baked something similar based on a recipe by Max Rumpolt (that I currently can't find), but since Rumpolt post-dates the Forme of Cury by about 400 years, I think the Sambocade is the earliest recipe I know.
As for the type of crust it uses, old bread was a very common ingredient in cooking. The Forme of Cury, Liber Cure Cocorum and many other cookbooks call for bread crusts or grated bread as ingredients to thicken sauces, or to use as a base for roasts or sweets. Bread was such an important staple food in Europe that it was ideally constantly available and never wasted (1).
Also, mixing bread crumbs with butter and sugar, then baking it is delicious. Let's not forget that taste was also a factor in food.
edit Since I didn't say, Sambocade is a 14th century, English recipe.