This all will go faster if you do all the chopping first: cut tips from the asparagus 1 1/2 inches from top and set aside for a good-looking garnish. Chop the rest of the asparagus stalks into 1/4-inch rounds. Slice the white and light green parts of the leeks. Chop the shallots into smallish pieces. Mince the garlic.
Melt the butter in a 4 to 5 quart pot on medium heat. Add the leeks and shallots and cook until translucent, about 5 minutes.
Stir in the garlic and cook a minute more.
Add the chopped asparagus (not the spear tips) to the onions. Sprinkle with salt and pepper to taste. Cook another 5 minutes.
Add the stock, water, thyme sprigs and bay leaf to the pot. Increase the heat to a boil, then reduce to a simmer. Cook, covered, until the asparagus are tender, about 10 minutes.
Toward the end of cooking, stir in the chopped parsley and arugula. You want it to wilt thoroughly.
While the soup is cooking, make a pretty garnish. Blanch the asparagus tips in a small pot of boiling, salted water, until the tips are just tender, about 2-4 minutes, depending on the size of the asparagus. Drain. Rinse with cool water to stop the cooking. Set aside.
Remove bay leaf and thyme sprigs. For a velvety texture, a Vitamix or similar high-speed blender is perfect. Use what you have, though: an immersion blender works, or a regular upright blender. If you use an ordinary blender, work in batches, fill no more than a third the blender bowl at a time, and hold down the lid while blending.
Wipe out your pan (so you don’t have any accidental lumps) and return the soup to the saucepan. Season with salt and pepper to taste, then stir in the vermouth and lemon juice. Taste again to check for balanced flavors.
At the end, drizzle in 2-3 Tablespoons of cream, stir well, and garnish with the reserved asparagus tips.
This soup is wonderful as a hot soup -- but it's also very refreshing served chilled.