Leave the potatoes whole. Do not peel. (Keeping the peels on keeps water out of the potatoes, very important.) Scrub them lightly. Place them in a medium saucepan and cover with water by about 3 inches. Boil until soft and tender but not breaking apart. About 30 - 40 min. Drain potatoes. Let cool to warm.
While the potatoes are boiling start the sauce. Drain ¾ of the juice off the tomatoes. Discard juice. Place tomatoes in a medium size shallow saucepan. Add salt, pepper and oregano. Stir occasionally and simmer uncovered for 20 min. (The rough texture on the tomatoes is most desirable). Stir in the olive oil. Cover and set sauce aside.
Peel the warm potatoes with a paring knife. Mash them with a potato ricer, potato masher or use your electric mixer. Stir the egg, butter and salt into the mashed potatoes. Add the flour a little at a time, and work into a smooth manageable dough. The mixture should hold its shape and be pliable but not sticky.
Dust your counter top with flour. Divide the dough in half. Roll each half into about a 1-inch thick rope, no more than 1-inch, and about 2 ft. long. Dust the counter top with a little more flour as you go if needed. (If it is easier for you, divide the dough into thirds, and roll shorter ropes, about 1 inch round and 1 ft. long – 10 gnocchi per rope.)
Using your dough scraper cut 20, 1-inch gnocchi per rope. Take a table fork and quickly but softly run it over the edge of each little gnocchi. This makes a nice little mark and will create a space for sauce to stick to it. Cut all the gnocchi before poaching.
Start 3 quarts of water boiling in a pot. Add 1 Tablespoon coarse salt. When the water is ready and boiling, poach the gnocchi in 4 batches (about 10 to each batch) by dropping them into the boiling water. Leave enough room for them to move without touching until they rise to the surface and roll over. This will take only about 2 min. They cook quickly.
When the gnocchi rise to the surface, carefully remove them with a slotted spoon to drain. Warm the sauce and serve immediately. Pass the Parmesan.
Plan Ahead: Make the gnocchi ahead of time, freeze or warm up in microwave. Make sauce ahead of time. Re-heat. Purchase fresh gnocchi, eliminate one step.