Make the broth: Combine the hocks, the vegetables, bay leaf, peppercorns, and 8 cups of water in a medium stockpot and bring to a boil over high heat. Turn the heat to low, put the lid on, and cook at a nice steady simmer until the meat on the hocks is so tender it's almost falling off the bone, 4 to 5 hours. Carefully remove the hocks and put them in a big bowl. Strain the cooking liquid through a sieve into the bowl with the hocks, and discard the vegetables and aromatics. Let the hocks cool in the liquid.
When the hocks are cool enough to handle, pull off the meat into bite-sized chunks. Discard any bones and any hard bits, and some of the fat, but don't throw away the skin — I add the skin to the soup in thin slices along with the chunks of ham. You don't have to, but I like the way it goes sticky in the soup. You can keep the stock and meat (moistened with a splash of stock) in separate airtight containers in the fridge for up to two days. Gently warm the meat and skin before proceeding with the recipe. This recipe requires only 4 cups of stock — you may freeze the leftover stock for up to a month for your next batch of soup.
Make the soup: Put the butter in a large pot that has a lid and set it over medium heat. Once the butter starts to froth, add the onions, carrots, and salt and stir. Cover the pot and cook, stirring every now and again, until the onions are soft and creamy (but not colored) and the carrots are tender but firm, about 15 minutes.
Add the wine and bring it to a boil (turn the heat up, if you need to). Let the wine boil until it's all but gone, about five minutes. Add the mint and 4 cups of the ham broth and bring the liquid to a boil, then add the peas. (At this point, the carrots will have bobbed to the top. I like to pick out most of the carrot chunks before pureeing the soup, then add them back after. That way, you can nibble on them in the soup, getting that bit of texture.) Cook at a simmer until the peas are warmed though and tender, about 5 minutes.
Blend the pot's contents, in batches, until smooth. Return all the bright-green pea puree to the large pot, add the ham pieces and carrots, and cook at a very gentle simmer for about 5 minutes, just to let the flavors mingle and heat the ham. Have a taste, and season with salt. How much you need to add will depend on how salty the ham hocks are.
Add a generous drizzle of olive oil, several twists of black pepper, and the torn mint leaves. Then add the crème fraiche in little blobs here and there, so everyone will get a bit. Serve the soup in the pot, with little bowls alongside.