Preheat oven to 425°F.
Separately wrap red and golden beets tightly in double layers of foil and roast in middle of oven until tender, 1 to 1 1/2 hours. Unwrap beets. While beets are cooling slightly, whisk together shallot, lemon juice, salt, and pepper in a small bowl, then add oil in a stream, whisking. When beets are cool enough to handle, slip off and discard skins. Separately cut red and golden beets into 1/4-inch dice and put in separate bowls. Add 2 1/2 tablespoons dressing to each bowl and toss to coat.
Place cookie cutter in center of 1 of 8 salad plates. Put one eighth of red beets in cutter and pack down with your fingertips. Crumble 2 teaspoons goat cheese on top, then one eighth of golden beets, packing them down. Gently lift cutter up and away from stack. Top with smoked salmon. Make 7 more servings in same manner. Drizzle each plate with 1 teaspoon dressing and scatter with some pistachios.
Toss mâche with just enough remaining dressing to coat and gently mound on top of smoked salmon. Enjoy!