Every cook must know how to make mashed potatoes. There are some critical steps: Start with a starchy potato, like Yukon gold or russet, not a waxy one, like red skin. Cut the potatoes into large pieces (at least 2 inches) so they don’t get waterlogged and become mealy. Start boiling them in cold water so the starches cook slowly and become creamy. Paddle the potatoes in a stand mixer to give them a fluffy texture.
For the potatoes: Place the potatoes in a 6- to 8-quart pot with a lid. Cover with cold water until the water is 1 to 2 inches above the potatoes. Add a heaping tablespoon of salt and bring to a boil, then reduce the heat to a medium simmer. Cover partially and cook until fork-tender, 10 to 15 minutes. (This is a good time to start the gravy.) Drain in a colander then return the potatoes to the pot and place over the lowest heat setting.
Combine the half-and-half and heavy cream in a microwave-safe container and microwave until steaming, about 45 seconds.
Transfer the potato mixture to the bowl of a stand mixer fitted with the paddle attachment. Paddle the potatoes on the lowest setting until they start to break apart, about 30 seconds. Add the warm cream mixture, then one piece of butter at a time and mix after each addition until just combined. Remove from the mixer, season with a generous pinch of salt, then taste and adjust as necessary.
For the gravy: Heat a 1-quart saucepan over medium-high heat until very hot, 1 to 2 minutes. Stir in the wine and bring to a boil over high heat, stirring constantly, until the wine reduces by 75%. Pour the wine into a small ramekin or heatproof bowl and reserve.
Melt the butter in the same saucepan over medium-high heat. Add the shallots and cook, stirring, until light to medium brown, 1 to 2 minutes. Stir in the flour until a roux (thick paste) forms. Reduce the heat to medium and cook the roux until smooth and medium blonde in color, about 1 minute.
Whisk in the reduced wine and 1 1/2 cups of the cold beef stock until the roux and the stock are completely incorporated. Increase the heat to high and bring to a boil. Cook until the gravy coats the back of a wooden spoon, whisking in up to 1/2 cup more stock for a thinner consistency, 2 to 3 minutes. Season with salt and pepper then remove from the heat.