In a large skillet, over medium heat, cook the bacon until crispy, 4 to 5 minutes. Remove the bacon, drain on a paper towel, and set aside. Remove all but 1 tablespoon of the grease, then add 1 tablespoon of the butter, the celery hearts, pepper, and 1/4 teaspoon of the salt. Cook for 2 minutes, tossing to coat well. Add the chicken broth, cover, and simmer until the celery is tender, about 20 minutes. Remove from heat. Finely chop parsley, garlic clove, and oregano, add to celery, and toss well. Meanwhile, bring 4 cups of water to a boil. Slowly whisk in the polenta. Add the remaining salt and cook, stirring frequently, for 5 minutes. Stir in 1/2 cup of the Parmesan and the remaining butter and mix well.
Divide the polenta among 4 bowls and top each with some celery and bacon. Spoon the cooking liquid on top and sprinkle with the remaining Parmesan.