Rinse the peas. Place the split peas in a fine-mesh strainer and rinse under cold running water until the water runs clear. Pick out any discoloured peas or debris. No soaking is necessary — yellow split peas break down fully with dry cooking.
Simmer with the onion. Combine the rinsed peas and the roughly chopped onion in a medium saucepan. Add 1 litre of cold water and bring to a boil over medium-high heat. As it comes to a boil, grey-white foam will rise to the surface — skim it off carefully with a spoon. Once skimmed, reduce the heat to a gentle simmer and cook uncovered, stirring occasionally, for 45–55 minutes, until the peas fully dissolve into a thick, porridge-like consistency. If the water evaporates too quickly before the peas are completely soft, add more hot water in small additions, 50ml at a time.
Blend in the olive oil. Remove the pan from the heat. Using an immersion blender — or working in batches with a regular blender — purée the mixture until completely smooth. With the motor running, stream in the warm olive oil in a thin, steady drizzle to emulsify it into the purée. Taste and season with salt and black pepper. If the fava is stiffer than you like, blend in hot water one tablespoon at a time; it will firm slightly as it cools.
Prepare the toppings. Thinly slice the red onion into rings, place in a small bowl with the lemon juice and a pinch of salt, and set aside for 5 minutes. Rinse the capers and pat dry.
Plate and serve. Spread the warm fava onto a shallow plate or wide bowl, smoothing the surface with the back of a spoon. Arrange the capers and marinated red onion on top, drizzle generously with extra-virgin olive oil, and finish with dried oregano, a final squeeze of lemon juice, and parsley if using. Serve warm or at room temperature.