Make the vinaigrette: Combine the minced shallot and vinegar in a small bowl. Add salt and pepper and whisk well. Add the horseradish and mustard and whisk to combine again. Drizzle the oil into the horseradish/vinegar/shallot mixture slowly as you whisk it vigorously until thoroughly combined. Alternatively, you can throw the whole works except the shallots into the blender and put it on high for a few seconds and add the shallots to the mix after. Set aside.

Place the potatoes in a medium saucepan and cover with cold water by about an inch or two. Bring to a boil over medium-high heat and cook until potatoes yield to a paring knife easily, but not so easily that they mush. About 12 minutes. Remove the potatoes with a slotted spoon and set aside. Place the beet roots in the boiling water and cook until you achieve the same knife-yielding tenderness of the potatoes. This should take longer, about 20 minutes. Remove with a slotted spoon and peel as soon as you are able, without burning your fingers of course. Cut potatoes and beets into small wedges and set aside in separate bowls.

Peel and pit the avocado. Slice it into 1/4 inch thick pieces and lay onto your serving plate. Squeeze the lemon over top of the avocado slices and season with salt and pepper. Drizzle a bit of the dressing here too. Arrange the beets on top of the avocado. Drizzle some of the dressing on top. Sprinkle with salt, pepper and some of the chives. Arrange the potatoes on top of the beets and drizzle with more dressing, salt, pepper and the remaining chives. Scatter the capers on top. Serve the avocado salad.