Bring potato to a boil in a pot of salted water. Cook until knife-tender, about 6 minutes; drain.
Grate beets and carrots on the large holes of a box grater into a large bowl (or shred in a food processor). Stir in chickpeas, potato, flour, and 1 teaspoon salt. Season with pepper.
In a large nonstick straight-sided skillet, heat 1 tablespoon oil over medium heat. Add beet mixture, and firmly press evenly into pan using a flexible spatula. Cook, undisturbed, 10 minutes. Remove pan from heat, cover with a serving platter, and carefully invert pancake onto platter. Slide pancake back into pan, and cook until crisp and brown around edge, 10 to 12 minutes. Invert pancake onto platter.
In a medium bowl, toss salad greens with remaining 2 tablespoons oil and the lemon juice. Season with salt and pepper. Top pancake with salad and goat cheese, and cut into wedges.
Sweet and earthy, this galette tastes even better when topped with a tangy, peppery mix of goat cheese and greens.