Here is my adaptation of a dish served to me years ago at a restaurant in Belden Alley called Plouf. It’s best to instruct the eaters to scoop potatoes onto the mussel shell to eat with the mussel and its butter. Yum!
In a large saucepan combine the potatoes, peeled and cut into 1-1/2 to 2 inch pieces, and the garlic with enough salted cold water to cover the potatoes by half an inch. Simmer them, covered, for 15 to 25 minutes, or until they are tender but not falling apart. Drain the potatoes and force them through a food mill fitted with the medium disk or a potato ricer into the saucepan. Stir in the cream (or olive oil) and salt and pepper to taste. Spread the mixture on two 10-inch round shallow flameproof platters.
Put the cleaned mussels in a large pot with the shallots and vermouth (or white wine.) Cover and place over high heat. Steam the mussels until they just open, about 5 minutes. Discard half of each shell, loosen the mussels and place them on the round end of the other half of the shell and embed these decoratively in the mashed potatoes.
Cream together the butter, parsley, and garlic. Put a large dab of the flavored butter on each mussel. Just before serving, run the mussels under the broiler for about 3 to 4 minutes, until bubbling.