Pat the venison cubes completely dry with paper towels. Season generously with salt and pepper, then toss in the flour until lightly coated. Shake off excess and set aside on a plate.
Heat 2 tablespoons olive oil in your Dutch oven over medium-high heat until shimmering. Add venison in a single layer, working in batches if needed. Sear each side for 2-3 minutes until deeply browned, then transfer to a plate and set aside.
In the same pot, add the bacon lardons and cook over medium heat until crisp, about 5 minutes. Add remaining olive oil with onion, carrots, and celery. Cook 6-8 minutes until softened. Add cremini mushrooms and cook 4 more minutes until they brown. Stir in garlic and cook 1 minute until fragrant.
Stir in tomato paste and cook 1 minute. Pour in red wine, scraping up browned bits from the bottom. Let wine simmer 3-4 minutes to cook off alcohol. Return browned venison and any juices to the pot. Add stock, soaked porcini, thyme, and bay leaves.
Bring mixture to a gentle simmer, cover with a lid, reduce heat to low, and braise for 1 hour 45 minutes to 2 hours, stirring occasionally, until venison shreds easily with a fork and sauce has thickened. Discard thyme sprigs and bay leaves.
Stir the redcurrant jelly into the hot filling until fully dissolved. Taste and adjust seasoning with more salt and pepper as needed. If sauce seems thin, simmer uncovered for 5-10 minutes to thicken. Let filling cool for about 20 minutes before assembling.
Preheat oven to 400°F. Pour cooled venison filling into a 9-inch deep pie dish, spreading evenly. Roll out puff pastry on a lightly floured surface until slightly larger than the dish. Drape over filling and trim excess overhang, leaving half an inch to tuck under. Crimp edges to seal and cut slits in the center for steam.
Brush the top of the pastry with beaten egg wash for a shiny, golden finish. Place pie on a baking sheet to catch drips and bake for 25-30 minutes until pastry is puffed and deep golden brown. Let rest for 10 minutes before slicing so filling can set.