Preheat the oven to 350 degrees.
In a small bowl, combine the sourdough pieces, olive oil, thyme, and salt and toss. Spread the croutons on a baking sheet and bake until golden, about 10 minutes, shaking the pan once or twice.
Meanwhile whisk the garlic, lemon juice, vinegar, mustard, and anchovies in the bottom of a large bowl. Gradually whisk in the olive oil. Season to taste with salt and pepper. Add the greens and toss to coat. Add the croutons and toss. Sprinkle with the cheese and serve.