Make the dressing: Combine shallot, brown sugar, dijon, paprika, salt, pepper, vinegar, and olive oil in a mason jar. Seal tightly, and shake until evenly combined. This can be done up to 3 days in advance. Keep refrigerated.
Roast the beets: Heat oven to 425°F. Place beets in a small baking dish. Rub with olive oil and season with salt and pepper. Cover with foil and roast until a knife easily goes through, about 40-50 minutes. Let cool then peel with a pairing knife and quarter.
Boil the eggs: Place eggs in a small saucepan. Cover with cold water and add a handful of salt. Bring to a boil. Remove from heat, cover and let sit 12 minutes. Drain and rinse with cold water until cool enough to handle. Peel and cut in half.
Arrange the salad: Arrange the lettuce in a large bowl or platter then top with quartered beets, halved eggs, sliced chicken, crumbled bacon, chopped avocado, chives, sliced radishes, and feta. Drizzle dressing over the top, season with salt and serve!