In a large bowl, add the pigs’ tails and enough cold water to cover by at least 3 inches. In a second large bowl, add the beans and enough cold water to cover by at least 4 inches. Cover the bowls and set aside to soak at room temperature for at least 8 hours or overnight.
Drain the tails, discarding their soaking liquid. Rinse the tails well, then add them to a 6-quart pressure cooker, along with just enough fresh, cold water to come ⅓ of the way up the sides of the pot. Bring to a boil over high heat, then cover the pressure cooker and add the steam valve. Once the pot reaches pressure, lower the heat to low and cook under pressure for 40 minutes. Turn off the heat and set aside without opening until the pot depressurizes naturally while you start the soup.
Drain the beans, discarding their soaking liquid, and set aside. To a large pot over medium heat, add the oil or pork fat. When the oil is hot, add the onion, garlic, and thyme and cook, stirring frequently, until fragrant and softened, 3–4 minutes. Add the sweet potatoes, the reserved beans, and 8 cups cold water. Bring to a boil, then lower the heat to simmer. Partially cover and cook, stirring occasionally, until the beans are nearly tender, 35–40 minutes.
Carefully open the pressure cooker, and use a slotted spoon to transfer the meat to the pot of beans. Strain and degrease the tail-cooking liquid; measure 1½ cups of the liquid and stir it into the beans as well, discarding the rest or reserving it for another use. (It will be quite salty and concentrated with collagen.) Bring the soup back up to a simmer and continue cooking until the beans and sweet potatoes are very tender and the tail meat has an almost “melting” consistency, 40–50 minutes more.
Meanwhile, make the dumplings: In a medium bowl, stir together the flour, cornmeal, sugar, and salt. Drizzle in ⅓ cup ice water, then use a fork or 2 knives to mix until a shaggy dough comes together. Lightly flour a clean work surface, transfer the dough out onto it, and knead until smooth and elastic, 6–8 minutes. Cover with a clean dish towel and set aside until the dough has softened and relaxed, about 15 minutes. Roll the dough out to an even, ½-inch thick snake, then use a heavy knife or bench scraper to cut the snake into ½-inch long dumplings. Add the dumplings to the soup along with the evaporated and milk and ⅔ cup of the brown sugar. Continue cooking, uncovered, until the dumplings are tender and the soup is thickened 40–50 minutes more. Season to taste with black pepper and additional brown sugar and salt (the soup should be quite sweet), and serve hot.