Make the red pepper sauce In a saucepan, combine the red pepper and tomato chunks with the 1 cup water. Bring to a boil over high heat, then reduce the heat to low. Cover and boil gently for 8 minutes; strain the puree through a food mill.
Return the puree to the saucepan and stir in the salt, pepper, and dissolved cornstarch. Bring to a boil, then add the butter and oil and emulsify with a hand blender for 10 seconds. Keep the sauce warm while you make the dumplings.
Make the dumplings Bring 3 quarts of water to a simmer in a pot. Meanwhile, in a food processor, combine the ricotta, salt, pepper, egg, Parmesan, and flour and process for 10 seconds. Transfer the dumpling batter to a bowl and mix in the chives.
Cook the dumplings in two batches Working as quickly as possible, drop the batter, about 1 tablespoon at a time, into the hot water. Let the dumplings cook at a low simmer (180°F to 190°F) for 10 minutes; the dumplings will rise to the surface of the water as they cook. Remove the dumplings with a slotted spoon, drain well, and arrange on a platter. Keep warm while you cook the remaining dumplings.
Arrange the dumplings on a platter or individual plates. Spoon on the warm red pepper sauce and sprinkle with grated Parmesan.
