Heat 1 Tbs. of olive oil in a skillet and sauté the onion, cumin and allspice over medium heat for 3 to 4 minutes. Set aside in a bowl.
Cook the beets in plenty of water until three-quarters cooked (roughly 20 minutes). Drain, cool and peel the beets. Grate coarsely on a box grater (you might want to use rubber gloves to do this!).
In a medium nonstick pan, bring the milk to the boil, then reduce to a simmer. Little by little, whisk in the chickpea flour until you have a smooth paste. Keep the mixture moving to avoid lumps. Season, then add the remaining olive oil and cook over low heat for 8 minutes, stirring all the time with a wooden spoon. Like cream puff dough, as the mixture heats, it will come away from the sides of the pan and shape into a ball.
Cool the ball of paste, then mix in the sautéed onions, chickpeas, lemon juice and grated beets. Using your hands, mold the mixture into golf-ball-sized balls and arrange on a baking sheet. Refrigerate for a couple of hours.
In a large saucepan, heat 2 inches of oil to 350°F. Carefully place the falafel into the oil and cook for 3-4 minutes, until tinged golden brown. Remove with a slotted spoon and place on paper towels to drain.