Remove the woody part of the asparagus stalk. Peel the remaining asparagus stalk and discard all peelings. Cut the tips and reserve them. Cut the remainder stalk into 1-inch length.
In the pan of your pressure cooker, add 1 tablespoon butter and add the stalks chunks of the asparagus and the green part of the leeks. Sauté for 3 minutes and then cover with 3 cups of water and 4 cups of chicken stock. Add tarragon spring. Follow your pressure cooker instruction, close the lid, and set the timer for 20 minutes.
Meanwhile in a small saucepan, bring 1 ¼ cup of lightly salted water to a boil. Stir in the rice and cover. Lower the heat and simmer for 13 minutes, or until the rice is almost done. Keep aside when done.
In a large frying pan melt the butter over medium-high heat. Add the whites of the leeks and cook, stirring frequently, for 5 minutes or until they are nicely brown.
Once the asparagus and leek soup in the pressure cooker is done, remove a ladle of the soup and place it in the skillet with the leeks to scrape up any brown bits.
Transfer the content of the soup pot to a large blender, add the caramelized leek and puree them together well.
Return them to the pan, add the cooked rice to it. Add the asparagus tips return to a boil. Lower the heat: simmer for 2 minutes. Season with salt and pepper.