Melt butter in a large skillet over medium heat. Stir in onion, mushrooms, and brown sugar. Cook and stir until onion has cooked to a deep brown color, about 10 minutes. Remove caramelized onions to a plate and set aside.

Meanwhile, fill a large pot with lightly salted water and bring to a rolling boil over high heat. Once water is boiling, stir in gnocchi and return to a boil. Cook until gnocchi floats to the top, about 3 minutes; drain and keep warm.

Place the same skillet over medium-high heat and add shrimp. Cook and stir until shrimp have turned pink on the outside and are no longer translucent in the center, about 5 minutes. Add caramelized onions, sour cream, paprika, and saffron. Cook and stir until mixture begins to simmer, then gently fold in cooked gnocchi. Season with salt.