For the lemon sauce: Heat a nonstick pan over high heat, place the lemon halves in the pan cut-sides down and cook until golden brown, about 2 minutes. Remove and let cool slightly.
Whisk together the honey, mustard and horseradish in a medium bowl, squeeze in the juice from the seared lemon and add the dill, mint and salt and pepper to taste. Let sit at room temperature for at least 30 minutes to allow the flavors to meld.
For the potato salad: Cover the potatoes with cold, salted water and bring to a boil. Lower to a simmer and continue to cook until the potatoes are fork-tender, 12 to 15 minutes. Drain and slice the potatoes into 1/4-inch-thick rounds.
Heat the oil in a large saute pan over medium heat. Add the bacon and cook until golden brown and slightly crisp, about 8 minutes. Remove the bacon with a slotted spoon to a plate lined with paper towels.
Add the shallots to the pan and cook until soft. Add the vinegar and sugar and cook until the sugar is dissolved. Remove from the heat, whisk in the mustard, add the potatoes and cornichon and gently mix until combined. Transfer to a platter and top with the crisp bacon.
For the beet reduction: Bring the beet juice to a boil in a small nonreactive pan over high heat. Cook until thickened and reduced to about 1/2 cup, about 12 minutes. Add the honey and season with the salt and pepper. Let cool slightly.
For the wiener schnitzel: Preheat the oven to 300 degrees F.
Spread the breadcrumbs out on a baking sheet and bake until pale and dried, about 12 minutes.
Increase the oven temperature to 350 degrees F. Set 2 baking racks on 2 baking sheets.
Put the flour in a baking dish and season with salt and pepper. Put 3 of the eggs in another baking dish with a few tablespoons of water, whisk until smooth and season with salt and pepper. Put the breadcrumbs in a third baking dish and season with salt and pepper.
Season the veal on both sides with salt and pepper. Dredge in the flour and tap off the excess. Dredge in the eggs, letting the excess run off. Dredge in the breadcrumbs and tap off the excess. Place on one of the baking racks.
Heat the oil in a large saute pan over medium heat until it begins to shimmer. Cook the veal, one cutlet at a time, until golden brown, about 4 minutes on each side. Place on the second baking rack and finish cooking in the oven for 5 minutes.
Melt the butter over medium heat in a 12-inch straight-sided skillet. Crack the remaining 4 eggs into the pan, season the tops with salt and pepper and cook until the whites are set and the yolks are slightly firm, about 2 minutes. Gently flip and continue cooking until the other side is just slightly set, about 30 seconds.
To serve: Divide the wiener schnitzel among large plates, drizzle with the lemon sauce, top each with a fried egg and drizzle with the beet reduction. Serve the potato salad on the side.