Loosely wrap foil around garlic, folding foil edges securely. Roast until garlic has softened, about 40 minutes, then transfer to plate. Open carefully and discard foil; let garlic cool.
Separate garlic into cloves. Squeeze soft garlic from each clove into a small bowl; set aside.
To make aioli: In a blender, pulse garlic, mayonnaise, tarragon, 1/4 teaspoon salt, olive oil and pepper to combine. Refrigerate until ready to use. (Note: Aioli can be made 1 day ahead and will keep 5 days in the refrigerator.)
In a large skillet, bring 2 quarts water and remaining 1/2 teaspoon salt to a boil. Add asparagus and cook until tender but still firm (a knife should easily pierce asparagus), 3 to 5 minutes. Drain and serve immediately with 1/2 cup of aioli. (Asparagus may also be chilled.)