Wrap garlic cloves with 1 tablespoon of the butter loosely in foil; fold edges to seal. Bake in a 400º oven until soft when pressed, 30 to 40 minutes.
Meanwhile, scrub and pull beards off mussels.
In a 4- to 5-quart pan over medium heat, combine remaining 2 tablespoons butter, thyme leaves, and 1/2 teaspoon salt. Squeeze soft garlic cloves from peels into pan. Stir until shallots are limp, 3 minutes.
Increase heat to high and add verjus; when boiling, add mussels, cover, and cook until shells have opened 2 to 3 minutes.
Sprinkle with parsley, pepper, and more salt to taste. Spoon mussels and liquid into wide, shallow bowls.