How to Make It
Preheat the oven to 375°. Brush the beets with oil and season with salt. Wrap the beets in foil and bake for 1 hour, until tender when pierced with a knife. Let cool slightly, then peel and cut into 1/2-inch wedges.
In a saucepan, cover the eggs with water and bring to a vigorous boil. Cover the saucepan, remove from the heat and let stand for 10 minutes. Drain and shake the pan to crack the eggs, then cool under cold running water and peel. Quarter 2 of the eggs lengthwise. Halve the remaining 2 eggs and scoop out the yolks. Press the 2 yolks through a fine sieve into a bowl. Finely chop the 2 egg whites.
Whisk the cream into the egg yolks until smooth, then whisk in the vinegar, mustard powder and brown sugar. Stir in half of the chopped egg whites and season with salt.
Pile the lettuce and arugula on a platter. Arrange the beets, tomatoes, cucumber, radishes and quartered eggs on top. Garnish with the scallions, parsley and remaining chopped egg whites; serve with the dressing.