Slice leeks in half lengthwise and rinse any grit or sand out of them. Place the leeks and carrots in a casserole dish. Drizzle two tablespoons of olive oil over the vegetables and lightly toss to coat. Sprinkle with rosemary, thyme, salt and pepper. Add two tablespoons vegetable broth and cover with lid or tight fitting aluminum foil.
Braise for 40-45 minutes, until vegetables are tender.
Meanwhile, place the bread in a mini prep food processor and pulse until you have fine breadcrumbs. Transfer breadcrumbs to a small bowl and add remaining tablespoon of olive oil. Toss to coat. Arrange breadcrumbs in a single layer and bake for 5-8 minutes until crisp. Set aside.
Transfer the vegetables to a serving platter and spoon any braising liquid over them. Sprinkle with breadcrumbs and serve.