Rinse the split peas under cold water until the water runs clear, picking out any small stones or debris.
Add the peas to a 3-quart / 2.8 L saucepan with the water, bay leaf, halved onion, and garlic cloves. Bring to a boil over high heat, then skim off any foam that rises to the top.
Reduce the heat to low and simmer uncovered for 40 to 45 minutes, stirring now and then, until the peas are fully soft and mash easily against the side of the pot. Add a little more water if the pot looks dry before the peas are done.
While the peas cook, heat 2 tablespoons olive oil in a 10-inch / 26 cm skillet over medium-low heat. Add the sliced onions and a pinch of salt.
Cook the onions for 25 to 30 minutes, stirring every few minutes, until deep golden brown and soft. Stir in the honey and vinegar for the last 2 minutes, then take the pan off the heat.
Once the peas are tender, remove the bay leaf. Drain off most of the liquid, keeping about 1 cup / 240 ml of the cooking water aside.
Mash the peas, onion, and garlic right in the pot with a potato masher, or blend with an immersion blender, until fairly smooth. Drizzle in the remaining olive oil gradually as you mash.
If the puree feels too thick, stir in a little of the reserved cooking water until it's thick but spreadable, like hummus. Stir in the lemon juice and season with salt to taste.
Spread the warm fava onto plates or a shallow serving dish. Top with the caramelized onions, a drizzle of olive oil, and a sprinkle of oregano or dill. Serve with lemon wedges.