Cover the potatoes by 1 inch with cold water, add 2 tsp salt and bring to a gentle boil.
Cook until a paring knife slides through a chunk without resistance, 15 to 18 minutes, then drain thoroughly and return the potatoes to the hot pot for 1 minute.
Mash the dry potatoes, then fold in 3 tbsp butter, the warm milk, sour cream and 1/4 tsp pepper just until smooth.
Melt the remaining 3 tbsp butter in a skillet over medium heat and cook the shallot until soft, 2 minutes.
Whisk in the flour and cook, whisking, until the roux turns the color of peanut butter, 3 to 4 minutes.
Slowly whisk in the broth and Worcestershire and simmer until glossy and thick enough to coat a spoon, 5 minutes.
Season the gravy with the remaining 1/2 tsp salt and 1/4 tsp pepper and spoon it over the hot potatoes.