Because mustard greens are on the cusp of chewy, it's a good idea to let the salad stand for a few minutes once you've tossed everything together. This will give the leaves a chance to soften.
Preheat oven to 400 degrees. Heat 1/4 cup plus 2 tablespoons oil in a small saucepan over low heat. Cook anchovies and garlic until anchovies are broken up and garlic is pale golden, 12 to 15 minutes. Pour mixture into a large bowl.
Meanwhile, toast bread on a rimmed baking sheet until golden brown, 8 to 10 minutes. Add to bowl with anchovy oil, and toss to coat.
Add mustard greens, lemon juice, and remaining tablespoon oil to bowl, and toss to coat. Season with salt and pepper. Let stand for 5 minutes before serving.