For the crab cakes: In a large bowl, mix together all of the ingredients except for the canola oil. Shape into 4 equal patties.
Heat the oil in a large skillet over medium heat until hot. Carefully add the crab cakes and fry until browned, 4 to 5 minutes. Carefully flip and fry on the other side until golden brown, about 4 minutes. Remove to a plate and keep warm.
For the fennel: Thinly slice the fennel on a Japanese mandoline. Transfer to a medium bowl; add the lemon juice and olive oil and toss well to combine. Set aside.
For the poached eggs: Bring a large saucepan of salted water to a boil and add the vinegar. Reduce the heat to a gentle simmer. While swirling the water with a slotted spoon, slowly add the 4 eggs and poach until the whites are solid and the yolks remains runny, 4 to 5 minutes. Carefully remove the eggs with the slotted spoon and drain on a paper towel-lined plate.
For the horseradish sauce: Combine all of the ingredients in a small bowl and stir well. Set aside.
Top each crab cake with some of the marinated fennel and 1 poached egg. Dollop with the horseradish sauce and serve right away.
Have 1 or 2 extra eggs on hand as a backup, just in case an egg falls apart during poaching.