Peel the carrots and grate on the fine holes of a box grater. Heat the olive oil in a medium skillet over medium-high heat. Add the carrots and a pinch of salt and cook, stirring constantly, until the carrots are softened but not brown and any water has evaporated, about 3 minutes.
Transfer to a bowl. Add the horseradish, Dijon mustard and sugar. Mix well, taste and add more salt, Dijon mustard and up to 2 more tablespoons horseradish if you like. Finish with a drizzle of olive oil and some torn mint leaves.
Be sure that the Dijon mustard is certified kosher for Passover.