In a 5- to 6-quart pan over high heat, bring 2 quarts water to a boil. Meanwhile, rinse asparagus; snap off and discard tough stem ends. Add asparagus to boiling water and cook until barely tender when pierced, 3 to 4 minutes. Drain and arrange on a platter or plates.
In a large bowl, mix 1 tablespoon olive oil and the vinegar. Add lettuce and tomatoes; mix gently to coat. Add salt and pepper to taste. Arrange salad alongside asparagus.
In the pan, over medium heat, stir garlic in remaining 6 tablespoons oil until limp, about 30 seconds. Add shelled beans and 2/3 cup water; cover and simmer over low heat until beans are very soft when pressed, 7 to 12 minutes. Mash with a potato masher. Remove from heat and push mixture through a wire strainer; discard skins.
Return bean paste to pan over low heat and stir until hot, 2 to 3 minutes. Remove from heat, add cheese, and stir until melted. Sauce should be thin enough to spoon over asparagus. If necessary, add a little hot water (2 to 4 tablespoons) to thin; do not return sauce to heat or cheese will become grainy. Add salt and pepper to taste.
Spoon warm sauce over warm or room-temperature asparagus.