Preheat the oven to 350°F (175°C) and line a large sheet tray with foil.
Carefully remove the stems from each mushroom, finely chop the stems, and set aside.
Wipe the mushrooms clean with a damp towel and place them hollow side down on the prepared sheet tray. Drizzle with 2 tbsp olive oil and a pinch of salt, rubbing to coat. Bake for 3-5 minutes, until the olive oil evaporates and the mushrooms are just beginning to brown.
In a large skillet over medium heat, add the remaining 1 tbsp olive oil. Add the leeks and a pinch of salt, sautéing until softened, about 5 minutes.
Add the sausage, breaking it up with a wooden spoon, and cook until browned and nearly cooked through, about 5 minutes.
Add the chopped mushroom stems and sauté for an additional 3-5 minutes, stirring occasionally. Stir in the garlic, red pepper flakes, and thyme, cooking until fragrant, about 1 minute.
Add the panko and cook until golden, about 2 minutes. Stir in the cream cheese until fully incorporated with the sausage mixture. Turn off the heat and fold in the Parmesan. Adjust seasoning as needed.
Turn the mushrooms hollow side up and fill each with a spoonful of the sausage mixture, pressing down to fill. (A slight mound on top is fine!)
Return the stuffed mushrooms to the oven and bake until the tops are golden brown, about 30 minutes. Serve immediately.