Place the rinsed yellow split peas, chopped onion, minced garlic, and 4 cups water in a large pot.
Bring the pot to a boil over medium-high heat, then immediately reduce the heat to low so the liquid maintains a gentle simmer.
Partially cover the pot and simmer about 45 minutes, stirring every 10–15 minutes, until the split peas are very soft and the mixture has thickened and starts to break apart when stirred.
Remove the pot from the heat and let it sit 3–5 minutes to cool slightly (this reduces splattering when blending).
Using an immersion blender, blend the mixture in the pot until smooth. If you do not have an immersion blender, carefully transfer the hot mixture to a countertop blender in batches and blend until smooth, venting the lid slightly and holding it with a towel to prevent pressure build-up.
Stir in the 1/4 cup olive oil and the juice of 1 lemon until fully incorporated.
Taste and season with salt and black pepper to your preference.
Transfer to a serving dish and serve warm or at room temperature. Garnish, if desired, with chopped onions, capers, and a drizzle of olive oil.