Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper.
Trim about 1/4 inch off the ends of the brioche loaf. Slice the brioche into four 2-inch slabs. Use two fingers to press a 1-inch-deep and 2-inch-wide well in the center of each slab, creating "nests" for your eggs. Crack the duck eggs into 4 small bowls.
Heat 2 tablespoons of the duck fat in a large nonstick skillet over high heat. When the fat is glistening, place 2 slices of the brioche well-side down in the skillet and fry the bread until it is golden, about 30 seconds. Flip the bread and toast the other side until golden, about 1 minute. Transfer the brioche to the baking sheet, well-side up. Repeat with the remaining 2 tablespoons duck fat and 2 slices of brioche. Gently slip an egg into each well and sprinkle with salt and pepper.
Bake the brioche until the egg whites are set and the yolks are jammy, 15 to 20 minutes.
While the eggs bake, prepare the slaw. Combine the Dijon and a pinch of salt and pepper in a medium bowl. Whisk in the vinegar. Whisking constantly, slowly add the oil.
Peel and slice the kohlrabi into matchsticks and add to the bowl with the dressing. Slice the apple into the matchsticks same size (leave the skin on) and add to the bowl. Toss the slaw with the vinaigrette and season to taste with salt and pepper.
Serve the brioche and eggs with the slaw on the side. Top with a sprinkle of parsley.