Place potatoes in a large pot and cover with water. Generously season with salt and bring to a boil of medium-high heat. Cover and cook for 20-25 minutes (depending on the size of your potatoes) or until tender. Drain potatoes and transfer to a large bowl. Using a potato masher or potato ricer, mash potatoes until crumbly. Add the melted butter and milk in varying intervals, stirring or mashing in-between each pour, until the potatoes are smooth and creamy. Season to taste with some salt and pepper. Add the egg and half of the cheddar cheese to the mash and whisk until combined. Cover the mashed potatoes, keeping the mixture warm, and set aside while you assemble the filling.

Melt butter in an oven-proof pot or dutch oven (4-6 quart) over medium heat. Once melted, add the veggies, and sauté until tender, about 2-3 minutes. Deglaze the pan with vermouth, add the whole grain mustard, and whisk to combine. Bring to a simmer, and, once the vermouth has reduced by half, add the flour and stir to coat the veggies. Add broth and milk to the pot, whisk to combine, and bring to a simmer. Simmer for 5-8 minutes, whisking frequently, or until the mixture has slightly thickened. Season the base to taste with salt and pepper.

Add the salmon, cod, shrimp, and scallops to the pot, and gently stir to combine. Cook for 3-5 minutes until seafood is just cooked. Remove the pot from the heat and evenly top with the mashed potatoes. Sprinkle the remaining cheese over the potato topping and place in the oven.

Broil until the potato-cheese topping is slightly golden. Remove from the oven and serve hot.