Drain the ricotta. Wrap the ricotta cheese in a cheese cloth, then place in a colander set over a large bowl. Weigh the ricotta down with something heavy to press out excess water. Allow to drain for 6 hours up to overnight in the fridge. Skipping this step will result in the gnudi falling apart during cooking.

In a large bowl, combine drained ricotta, ¼ cup Pecorino Romano, ¼ cup flour, 1 large egg yolk, 1 tsp Kosher salt, and ¼ tsp black pepper. Use a spatula to thoroughly mix.

Divide the mixture into halves, then roll each portion on flouted board into a log about 1-inch in diameter. Use a bench scraper to cut the gnudi into 1-inch long pieces. With lightly floured hands, gently roll each gnudi into a uniform ball, then place onto the floured baking sheet. Repeat with remaining gnudi. Refrigerate for at least 30 minutes.

While gnudi are chilling, purée tomatoes in blender. Pour puree through cheesecloth-lined colander set in bowl to collect the juice. Discard pulp and set juice aside at room temperature.

Bring large pot of salted water to a boil. Carefully lower the gnudi into water. They should float to the surface after a few minutes. Cooking time sb around 5-6 minutes; test by poking one with finger: it should feel soft but not liquid in the middle. Use a slotted spoon to transfer cooked gnudi to bowl with softened butter. Gently flip dumplings in the bowl until completely coated with butter.

Divide tomato water among 4 small bowls, then carefully drop buttered gnudi over liquid. Top with grated Parmesan and serve immediately.