Preheat oven to 375°F (190°C). Bring a large pot of water to a boil. Peel and roughly chop 2 lb Yukon Gold potatoes, then add to boiling water. Cover and cook until fork tender, about 15 minutes. When finished, add to a large bowl and mash, then mix in ½ cup full fat plain Greek yogurt, ¼ cup unsalted butter, and ¼ tsp salt. Set aside.
In a large oven-safe pot, heat 2 Tbsp olive oil over medium-high heat. Add 16 oz sliced mushrooms and cook until they lose their moisture and begin to turn golden brown, about 5 minutes. Add 1 medium yellow onion (sliced), 3 cloves garlic (minced), 4 medium carrots (diced), and 4 medium ribs celery (diced), continuing to cook until vegetables are slightly soft, about 7 to 10 minutes.
Stir in 2 Tbsp fresh herbs, 2 Tbsp tomato paste, 1 tsp fennel seed, and 1 tsp smoked paprika. Sprinkle in 3 Tbsp all-purpose flour and stir to evenly coat everything. Slowly drizzle in 2 cups vegetable broth, stirring as you add it. Heat the mixture until it thickens slightly. Stir in 1 14-oz can corn and 1 cup frozen peas.
Spoon the mashed potatoes over the filling in the pot, spreading it into an even layer (you can crimp the top with a fork to get little crispy edge pieces).
Bake, uncovered, for 20 minutes. If desired, cook under the broiler for a few minutes to help brown the surface of the mashed potatoes.
