Trim the ends of the leeks and cut in half lengthwise. Rinse thoroughly under running water being careful to get in between each layer of the leek (they can be very dirty). Transfer to a cutting board and cut into half inch pieces.
Melt 1 tablespoon butter in a large dutch oven over medium heat. Add a handful (about ½ cup) of attractive asparagus tips and 1 tablespoon of water. Cover and steam until asparagus tips are bright green and nearly tender, about 2 minutes. Uncover and sauté until completely tender, about 1 more minute, adjusting the heat as necessary to keep the butter from getting too dark on the bottom of the pot. If your asparagus tips are on the thicker side, this may take longer. Remove asparagus tips from the pot, sprinkle with coarse salt and reserve for garnish.
Add remaining 2 tablespoons butter to dutch oven. Once the butter has melted and foaming subsides, add leeks, thyme and 1 teaspoon salt. Saute leeks and thyme in butter until leeks are soft and beginning to brown, about 5-7 minutes.
Add flour and continue to cook, stirring constantly, for 1 minute.
Add remaining asparagus, broth and 2 cups water. Bring to a boil and reduce to a simmer. Cook until vegetables are just tender, about 5-7 minutes. (The time it takes will depend on the thickness of your asparagus spears. The thicker ones will take longer. The best way to know if they are tender is to test one.) Add frozen peas and remove from heat.
Puree soup in three batches in a blender until completely smooth. Return to pot. Add 2 tablespoons of lemon juice, cream and just a pinch or two of cayenne pepper. Season generously with salt and additional lemon juice to taste. Serve, garnishing each bowl with a few the reserved asparagus tips.