Begin by preparing your asparagus. Wash them thoroughly and trim off the tough ends. With the remaining stalks, preserve the soft tips of about 20 of your asparagus in a separate bowl.
In a large dutch oven, heat your butter until fragrant and bubbly on medium heat. Add in the diced onion and sautee for 2-3 minutes. Add the remainder of your asparagus stalks, save the preserved soft tips. Stir and allow the stalks to soften for 8 - 10 minutes. If the onions begin to stick to the bottom of the pan, add a little bit of olive oil.
Add two tablespoons of your flour to the mixture and stir to evenly coat the asparagus and onion mixture and to begin cooking through the flour. You will need to stir frequently for another 8 - 10 minutes to prevent sticking to the bottom of the pan.
Add your two cups of water and bring to a boil; simmer for another 8 - 10 minutes; feel free to add a cover or partial-cover to your dutch oven to ensure that the asparagus become soft.
In a separate pan, sautee the soft tips of the asparagus with olive oil and salt until just cooked. Set aside.
When you think the asparagus in the dutch oven are soft enough to puree, remove from heat, and add your milk. Puree in a small batches in a blender, and then return to your dutch oven. Simmer the soup in its pureed form for 3-4 minutes.