Add 1 cup of water and farro to a large pot. Bring to a boil over high heat. Reduce heat to medium-low, cover pot, and simmer 30–40 minutes, or until farro is al dente. Drain any remaining water and let cool.
While farro is cooking, create dressing by mixing shallot, yogurt, mint and 1 Tbsp. lemon juice in a small jar with lid. Add lid and shake well to incorporate. Season with salt and pepper to taste.
In a separate pot, bring 1 cup of water to a boil over medium-high heat. Place eggs into water one at a time using a slotted spoon. Reduce heat to medium-low and cook eggs 6–7 minutes. Transfer eggs to a small bowl of ice water and chill for 2 minutes. Gently crack and peel eggs; set aside.
Toss mixed greens, red onion, green beans, and halved grape tomatoes with 1 Tbsp. lemon juice in a medium bowl; season with salt and pepper. Add farro to bowl and toss to combine.
When ready to serve, plate salad. Cut soft-boiled eggs in half and add to plates. Drizzle salad with mint yogurt dressing.