Cover the mushrooms with the water and rehydrate for 15 minutes.
Peel the potatoes and cut into 1 inch pieces. Boil the potatoes in a large pot of lightly salted water until just tender, drain very well and pass through a food mill into a mixing bowl while still hot. Cover the bowl with cling film and put in a warm place until needed.
Meanwhile, swish the mushrooms to remove grit, then lift the mushrooms out of the water. Strain the liquid, discarding the last tablespoon of liquid with any grit. Reserve the mushrooms and liquid separately.
Sweat the shallots, (grate the garlic into the pan now), in 2 tablespoons of the butter until translucent, then add the mushrooms and sweat for a few minutes more.
Add the black trumpet liquid and reduce until the pan is nearly dry. Add the cream and salt, bring to a simmer. Turn down the heat and reduce for 5 minutes more on medium-low.
Pour the trumpet cream into a blender like a vitamix, or a food processor in a pinch. Puree until very smooth, scraping down the sides if needed. Drop in the butter at the end.
Pour the mushroom cream into the hot riced potatoes, combine, season to taste with salt. Adjust to your desired consistency by adding a splash of hot cream or sour cream if it's thick.
Notes
Using fresh mushrooms
You can make a simpler version with fresh mushrooms too. To do that, heat some butter in a large skillet over medium heat, sauté mushrooms, turning the heat to medium high. Season with salt and pepper