Season and brown the pork: Pat the pork dry and season with salt and pepper. Heat 2 tablespoons of olive oil in a large heavy-bottomed pot or Dutch oven set over medium-high heat. Brown the pork in batches until golden on all sides, then transfer to a plate.
Coat with flour: Return (all) the pork to the pot and sprinkle with ⅓ cup of flour. Stir well so the flour coats the pork, then cook for about 1 minute, stirring often. Remove the flour-coated pork from the pot and set aside.
Deglaze: Set the pot over medium heat and pour in ½ cup of white wine. Scrape up any browned bits from the bottom of the pot. Add 3 tablespoons of butter and stir until melted.
Cook the vegetables: Add the onion, carrots, and celery. Cook for 4 to 5 minutes, stirring often, until the onion begins to soften.
Add the mushrooms and aromatics: Stir in the mushrooms and cook for 3 minutes. Add the minced garlic, 1 teaspoon dried thyme, the fresh thyme, 1 teaspoon of paprika, 1 tablespoon of Dijon, and bay leaf. Stir for 30 seconds, just until fragrant.
Simmer: Pour in the broth and add the potatoes. Return the pork (and any juices) to the pot. Bring to a gentle simmer, then reduce heat to low, cover, and cook for 75 to 90 minutes, stirring occasionally, until the pork is fork-tender and the potatoes are cooked through.
Finish and serve: Remove the bay leaf. Taste and adjust seasoning with more salt and pepper as needed. Garnish with parsley, if desired, and serve hot.