Put the potatoes in a large pot and cover with cold water by about 1 inch. Add the tablespoon of salt. Bring to a boil, then lower the heat and simmer for 15 to 20 minutes, until a knife slides into the potatoes with no resistance.

While the potatoes cook, start the gravy. Melt the 3 tablespoons of butter in a large skillet or saucepan over medium heat. Add the mushrooms and onion and cook for 8 to 10 minutes, stirring now and then, until deeply browned. Stir in the garlic and thyme for 30 seconds.

Sprinkle the flour over the mushrooms and stir for 1 minute. Slowly whisk in the broth. Simmer, stirring often, for 6 to 8 minutes, until the gravy is thick enough to coat a spoon. Stir in the soy sauce and pepper, then taste and add salt if needed. Keep warm over low heat.

Drain the potatoes well and return them to the hot pot. Set the pot over low heat for 1 minute, shaking it once or twice, to steam off the extra moisture.

Take the pot off the heat. Mash the potatoes with the 6 tablespoons of butter until smooth. Stir in the warm milk a little at a time, then the sour cream, 1/2 teaspoon salt and the pepper. Stop as soon as the potatoes are creamy, since overmixing makes them gluey.

Spoon the mashed potatoes into a warm serving bowl, make a well in the top, and ladle over some of the gravy. Serve the rest of the gravy at the table.