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Remove the skin from the chicken legs and divide them at the joint. Sprinkle them generously with salt and pepper.
Heat the oil in a large, heavy-bottom pan, a pan large enough to hold the chicken pieces in one layer. Place the chicken parts in the pan and brown all over. Sprinkle the paprika on top and stir well to coat the chicken parts with it. Add the chicken stock, it should cover the meat almost entirely, but the chicken should not be swimming in liquid.
Chop the parsley and add half of it to the pan. Add some pepper, bring to a boil, lower the heat and cover the pan leaving a crack open. Cook for about 30 minutes (turning the chicken once in between) or until the meat is cooked through.
Start making the polenta about now if you want to serve this immediately.
By the end of the cooking time, the sauce should have reduced by half. If you want a thicker sauce, remove the lid from the pot during the last 5 minutes of the cooking time. I don't find it necessary, I like to have a runnier sauce to soak the polenta.
Grate the garlic cloves and add them to the pan together with the rest of the chopped parsley. Cook for 1 minute, remove from heat and season.
To make the polenta, heat the water in a large pan but do not let it come to a boil. Add the salt, stir to dissolve. Add the polenta in a slow stream while stirring all the time.
Bring to a boil, lower the heat, cover with a crack and cook, stirring often, according to the packet's instructions. Mine takes about 20 minutes, but that can differ greatly depending on the polenta you bought.
Once the polenta is ready, stir in the butter.
Serve as soon as the polenta is ready.