In a small bowl whisk together the yogurt, lemon zest and juice, mint, tahini, water, salt and pepper. Set aside.
Bring 6 cups of salted water to boil in large pot. Add barley and return to boil. Cook until tender, 20 minutes. Drain barley, transfer to bowl with the 2 teaspoons of mint and 2 tablespoons olive oil, tossing to combine. Cover to keep warm. Season with salt to taste.
While barley cooks, trim and halve carrots crosswise, then halve or quarter lengthwise to create uniformly sized pieces. Heat 2 tablespoons olive oil in large skillet over medium-high heat until just smoking. Add carrots and ½ teaspoon coriander and cook, stirring occasionally, until lightly charred and just tender, 5 to 7 minutes. Stir in snow peas, salt, cumin, cardamom, remaining ½ teaspoon coriander and cook until spotty brown, 3 to 5 minutes. Transfer to bowl with barley.
Using paper towel, wipe clean now-empty skillet. Add pine nuts and toast to golden brown, stirring constantly, about 2 minutes. Set aside.
Toss the barley roasted vegetable mixture to combine and pour onto platter or individual bowls. Drizzle tahini yogurt sauce all over, sprinkle with pine nuts, pomegranate seeds, mint leaves, and serve.