A pizza-sized cookie is... still just a cookie 
I'm not saying it's bad, it's just not dessert pizza.
I know, for certain, that while I'm sure a handful of NY pizzerias, over the years, have tried versions of dessert pizza, there is no pervasive history in the area. I'm also relatively certain that the Midwest has mostly steered clear of this phenomenon as well. On the left coast, they do some pretty wacky stuff, so I have no doubt that dessert pizza, to an extent, is a thing there, but I'm sure it's mostly Neapolitan inspired, and thus doesn't lay any claim to defining it.
Long windedness aside, dessert pizza is, at it's core, Neapolitan. Perhaps not Neapolitan in relation to Naples, but more in relation to Neapolitan style pizza as it is served here in the U.S. Based on Naple's infatuation with both pizza and Nutella, I think it's safe to say that someone there most likely had the idea first, and I know of one or two places there that go this route. On this side of the pond, though, I've never come across a Neapolitan pizzeria that didn't offer an least one dessert pie- which is almost always Nutella. The absolute ubiquitousness of the Nutella pie in domestic Neapolitan pizzerias is Dessert Pizza's greatest defining component.
Here's the most basic approach:
Form the pizza skin
Dock it (this is one of the only times Neapolitan pizza dough is docked) - you can alternately not dock it, let it blow up like a pita, and stuff it with fillings
Bake it untopped
Immediately out of the oven, spread a generous schmear of Nutella, sprinkle with powdered sugar (and possibly some whipped cream)
Serve
That's as bare bones as you get, and, as long as you're working with a Neapolitan capable oven/a sub 90 second bake time, the results may not match up with a quality baklava or french pastry, but it's not the worse dessert in the world. The residual heat from the pizza helps melt the Nutella a bit, and the resulting ooziness is pretty delightful.
One can build on this a bit. I have my clients give the baked skin a quick light brush of butter prior to the Nutella schmear. The butter takes the very lean dough and gives it a bit more richness. The bread can only absorb a very small amount of melted butter, though, so you want to use a light hand. If you're too excessive, the resulting pizza can get incredibly messy.
You can also go as crazy as you like with the toppings. This is pretty creative
https://www.pizzamaking.com/forum/index.php?topic=43129.msg431717#msg431717
Nutella, bacon, bananas, candied jalapenos, aka, the 'Flaming Elvis.'
At the end of the day, the most critical aspect, imo, is the oven. You're not going to see the same results with a longer baked pizza in a non Neapolitan capable oven. Fortunately, the number of consumer level Neapolitan capable ovens is growing, so this delicacy is much more obtainable for the home baker than in years past.
