Summer Vegetable Soup With Pistou

Ingredients for Soup:
– 1 pound cherry tomatoes, halved
– 1 ear of corn, kernels removed (approximately 1 cup)
– ¼ cup olive oil, with more for drizzling
– 1 teaspoon Diamond Crystal salt or ½ teaspoon Morton kosher salt, with more for seasoning
– Freshly ground pepper
– 5 large thyme sprigs (divided)
– 1 small fennel bulb, finely chopped (fronds removed)
– 2 medium leeks, sliced thinly after quartering lengthwise (using white and pale green parts only)
– 4 finely chopped garlic cloves
– 1 fresh or dried bay leaf
– 8 cups low-sodium vegetable or chicken broth
– ¾ cup pearl barley or farro
– 6 ounces green beans, cut into 2-inch pieces after trimming
– 1 medium zucchini, cut into ½-inch pieces after quartering lengthwise
– 1 bunch mature spinach, with stems removed and leaves thinly sliced (about 2 cups)

For Pistou and Assembly:
– 1 cup finely chopped basil
– 2 tablespoons finely chopped chives
– 2 tablespoons finely chopped mint
– 4 tablespoons (or more) olive oil
– 2 tablespoons finely grated Pecorino Romano cheese, with more for serving
– 1 teaspoon finely grated lemon zest