Combine warm water, sugar, and yeast. Let sit 5 to 10 minutes until foamy.
Add salt and 1 cup flour. Stir to combine. Add second cup of flour and stir 2 to 3 minutes.
Knead on a floured surface for 6 to 8 minutes until smooth. Place in an oiled bowl, cover, and let rise 45 minutes.
Divide dough into 2 pieces. Knead briefly into rounds. Cover and rest 15 minutes.
Blend gorgonzola, mascarpone, and cream cheese in a food processor until smooth. Set aside.
Preheat oven to 450 degrees Fahrenheit.
Roll one dough round into a 10-inch oval, about ¼ inch thick. Transfer to an oiled pan.
Top with tomato slices, brie chunks, spoonfuls of the cheese mixture, and prosciutto cotto.
Drizzle with olive oil and sprinkle with dried oregano.
Bake 15 to 18 minutes until edges are golden and cheese is bubbling. Rest 2 minutes before slicing.