Continue to let the farro cook until done, according to package directions, usually a total of 20 to 40 minutes. Drain farro.
As farro cooks, prepare the dressing: In a medium bowl, whisk together lemon juice, garlic and 1/4 teaspoon salt. Let sit for 1 minute, then whisk in oil, a few drops at a time, until emulsified. Whisk in tahini and set aside.
Position the rack underneath your broiler so that it's at least 4 inches away from the heating element; heat the broiler.
On a rimmed baking sheet, toss broccoli with olive oil and soy sauce, then spread the pieces out into an even layer. Broil until slightly charred in spots, 2 to 5 minutes, watching closely so that it doesn't burn all over (a few burned spots are OK). Let cool slightly, then toss with sesame oil and sesame seeds and cover to keep warm. (Alternately, you can also roast the broccoli at 450 degrees for 8 to 15 minutes instead of broiling.)
Toss cooked farro with 5 to 6 tablespoons tahini dressing to taste, a large pinch of salt and a drizzle of olive oil. Taste, and add salt and olive oil if needed.
To serve, divide farro among 4 serving bowls and drizzle with remaining dressing. Top with turnips, and sprinkle them with lemon juice and salt. Add broccoli and egg to the bowl and garnish with green onions and more sesame seeds. Serve immediately, with soy sauce, hot sauce and/or sliced chiles on the side.