Rinse the split peas in a fine-mesh sieve under cold water until the water runs mostly clear. Dice the onion and carrot, mince the garlic, and thinly slice the red onion for serving.
Heat the 2 tbsp olive oil in a 24 cm / 9.5-inch heavy-bottomed pot over medium heat. Add the onion, carrot and 1/2 tsp salt, and cook for 8 minutes, stirring, until soft. Add the garlic and oregano and cook for 1 minute, until fragrant.
Add the split peas, bay leaf and water. Bring to a boil over high heat, which takes about 7 minutes, and skim off any foam. Lower the heat, partly cover, and simmer for 40 to 45 minutes. Stir every 10 minutes and scrape the bottom, until the peas are very soft and fall apart when pressed with a spoon.
Remove the bay leaf. Blend about half the soup with an immersion blender in short pulses, so some whole peas remain. Take the pot off the heat, then stir in the lemon juice, the remaining 1 tsp salt and the pepper. Taste and add a splash of hot water if the soup is thicker than you like.
Ladle the soup into six bowls. Top each with red onion, capers, parsley and 1/2 tbsp extra virgin olive oil. Serve with a lemon wedge.