PREPARE THE VEGETABLES. Preheat the oven to 400ºF. Loosely wrap the carrots and beets in separate foil packets. Place the packets on a baking tray and roast for about 30 minutes, or until a sharp knife can be inserted easily: vegetables should be just tender, not mushy, with just the slightest bite to them. Depending on the size of the beets, you may need to remove the carrots first and let the beets continue to roast for another 10 to 15 minutes. Open the foil to and let the vegetables sit until they are cool enough to handle. Cut the carrots in half, then cut them into thin slices, about 1/8-inch thick. Cut the lay the slices flat on the cutting board and cut into thin strips. Cut strips crosswise into uniform 1/8-inch pieces. Transfer carrot pieces into a small bowl. Peel the beets with a paring knife: the skin should come off easily after roasting. Cut them into slices about 1/8-inch thick. Lay the slices flat then cut into thin strips. Cut strips crosswise into uniform 1/8-inch pieces. Transfer beets to a separate small bowl.
MAKE THE DRESSING. In a bowl mix together 2 tablespoons of the lemon juice, mayonnaise, pickle, shallot, capers, Dijon mustard, vinegar, and garlic. Taste and add a sprinkle of salt if desired.
PREPARE THE AVOCADO. Using a large knife, cut the avocado in half. Remove the pit by gently whacking it with the knife and twisting. Discard the pit. Holding the avocado in the skin, cut 1/4-inch pieces by slicing vertically then horizontally. Use a spoon to scoop the diced avocado into a small bowl. Gently stir in the remaining 1 tablespoon of lemon juice and a good sprinkle of salt.
ASSEMBLE AND SERVE. Place about 2 tablespoons of dressing into each of the two bowls with the carrots and beets. Stir to combine. Depending on the size of your ring mold, decide how many plates you will make. Choose plates several inches larger in diameter than your ring mold. Place the ring mold on the plate. Spoon the avocado into the bottom of the ring mold and use the spoon to gently press it down into a compact layer. Next, spoon in the beets and gently press down so that the layers are uniform. Finally, spoon in the carrots and press down again. Carefully lift the ring mold straight up to reveal the stacked vegetables. Repeat process for additional plates until ingredients are gone. Garnish with several long chives or chop and sprinkle them on the plate. Serve with toasted crostini or good quality crackers.