Make the oatmeal: In a medium saucepan, melt the butter over medium-high heat. Add the oatmeal and completely coat in the butter, toasting for 1 to 2 minutes. Pour in the stock and bring to a simmer. Cook, stirring occasionally, until tender and thickened, 10 to 12 minutes. Season with salt and keep warm on the stove.
Make the sausage: In a medium skillet, melt the butter over medium-high heat. Add the shallots and cook until lightly golden, 3 to 4 minutes. Add the soppressata and cook, 3 to 4 minutes more. Season with salt and keep warm.
Make the beans: Bring a pot of salted water to a boil. Add the beans and blanch until tender but still crisp, 1 minute. Drain the beans and transfer to a small bowl. Toss with the remaining bean ingredients and season with salt, then set aside.
Poach the eggs: Bring a pot of water to 180º and stir in the vinegar and salt. Crack each egg into a separate small bowl. Using a slotted spoon, swirl the water as you drop each egg into the pot. Cook, stirring as needed to prevent the eggs from sticking, until the whites are cooked but the yolks are still runny, 3 to 4 minutes. Using a slotted spoon, transfer the poached eggs to a paper towel-lined plate to drain.
Assemble the bowls: Divide the oatmeal between 4 bowls. Top each with the sausage, beans and a poached egg. Garnish with grated horseradish and serve.