In a large saucepan, combine 2 tablespoons of the olive oil, the onions, and garlic and sweat over medium heat until they’re translucent, 3 to 5 minutes. Add the farro and toast for 2 minutes, stirring frequently to make sure the onions and garlic don’t brown. Add the brown sugar, carrot juice, and 1 teaspoon salt and simmer over medium-low heat, stirring occasionally, until the farro is al dente—chewy but still firm, 20 to 30 minutes.
Add the diced carrots and continue to cook, stirring often, until the farro is cooked through and the carrots are tender, 10 to 15 minutes.
Taste the farro and season with salt, if needed, then drain in a colander (save any leftover cooking liquid in the refrigerator to use in a salad dressing). Spread the farro out on a large sheet pan and drizzle with the remaining 2 tablespoons olive oil, stirring to coat it well. Let it sit at room temperature until cool.
Place the cooled farro in a large bowl and toss with the lemon zest, lemon juice, shallot, radishes, and cilantro. (The salad can be made to this point up to 1 day ahead and refrigerated.)
When ready to serve, add the diced avocado, gently stir, and finish with the sesame seeds and a few twists of black pepper.
