To make the mashed potatoes: In a large pot, add 8 cups of water and 1 teaspoon of the salt; bring to a boil over high heat.
Rinse the potatoes under cold water to remove the starch. Drain and add to the boiling water. Cook until fork tender, 12 to 15 minutes; drain and return the potatoes to the pot.
While the potatoes are cooking, melt the butter in a small saucepan over low heat. Add the garlic and cook for 5 minutes, or until soft, taking care not to burn the garlic.
Use a potato masher to lightly break down the potatoes until mostly smooth. Gently fold in the garlic butter. Gradually stir in the half-and-half and season with remaining 3/4 teaspoon salt and the pepper. Keep the potatoes warm while you make the shepard’s pie filling.
Preheat oven to 375 degrees F.
To make the filling: In a large cast-iron or heavy-bottomed skillet, heat the olive oil over medium heat. Add the onion and sauté for 2 to 3 minutes, just until the onions start to soften. Add the mushrooms, carrots, rosemary, and thyme. Sauté for 5 to 6 minutes, or until the carrots are just starting to soften. Add the garlic and cook for 1 to 2 minutes, or until the garlic is fragrant.
Add the tomato paste and flour to the skillet and stir. Cook for 1 to 2 minutes, stirring constantly to cook off the taste of the flour. Add the wine and deglaze the skillet, stirring to scrape up any brown bits from the bottom of the skillet. Add the Worcestershire and broth, stirring until the gravy starts to form. Simmer over medium-low heat for 5 minutes, or until the gravy starts to thicken.
Remove the rosemary and thyme. Stir in the frozen peas and corn. Season with salt and pepper.
If you are using a cast-iron or other ovenproof pan, you can spread the reserved mashed potatoes right over the vegetable filling. Otherwise, transfer the filling to a 9-inch square or oval baking dish. Spread the mashed potatoes over the top of the filling.
Bake for 15 to 20 minutes, or until the mashed potatoes are starting to brown and the filling is bubbly. Serve immediately.