Put beets, vinegar, 2 tablespoons olive oil, and garlic into a saucepan. Add water just to cover. Bring to a boil, then reduce to medium and cook uncovered until fork tender, turning as liquid reduces (30-40 minutes). Remove beets and set aside. Remove and reserve garlic clove. Boil remaining cooking liquid until reduced to a 1/4 cup (about 8 minutes). Cool slightly.
Cut beets into 1/2 inch slices (no need to peel - the skin gets pleasantly crispy). Put between two sheets of wax paper and thump with something heavy (a can or rolling pin) to flatten slightly. Lightly salt both sides. Heat remaining 3 tablespoons of olive oil in a large fry pan over medium heat until shimmering. Add beets, making sure you have room to flip them. Cook until you have a brown crust (approx. 5 minutes), then flip and cook for another 3 minutes or so. Remove to a plate.
Grate hard boiled egg on largest holes of grater and set aside.
Mash garlic clove and add to reduced cooking liquid then whisk in dijon mustard, lemon juice, salt and pepper,
Put lettuces on a platter and top with beets and egg. Drizzle dressing over and serve.