Step 1

Heat grill to medium-high. Once hot, clean and oil grill grates with canola oil.

Step 2

Toss mushrooms with 1 tablespoon olive oil. Season with salt and pepper. Brush bread with 1 tablespoon olive oil. Grill mushrooms and bread, covered and turning often, until mushrooms are tender and slightly charred and bread is toasted, 3 to 8 minutes for mushrooms and 1 to 2 minutes for bread. Transfer to cutting board; rub bread with 1 garlic clove. Cut mushrooms into pieces. Tear bread into chunks.

Step 3

Finely grate remaining garlic clove into a bowl. Add anchovies, vinegar, and remaining 2 tablespoons olive oil and stir to combine. Cut 1/4 inch off top of tomato and finely grate cut side into bowl just until you reach skin; discard skin. Season with salt and pepper. Add torn bread, arugula, and mushrooms and toss to combine.

Step 4

Arrange arugula mixture on a platter. Top with chicken and any remaining vinaigrette in bowl.