In a medium saucepan, heat 1 tablespoon of the butter, then add the leeks and cook over medium-low heat for 10 minutes, stirring occasionally. Increase the heat to medium. Add the asparagus and 3/4 cup of the chicken broth and cook, uncovered, for 4 minutes. Remove from the heat.
Melt the remaining 1 tablespoon of butter in a small nonstick skillet over medium-high heat. When the butter bubbles, add the flour and stir vigorously with a wooden spoon for 1 minute, then reduce the heat to medium and continue stirring for 1 minute more. Gradually whisk in 1 cup more of the chicken broth. Bring the mixture to a simmer, whisking until smooth and thickened. Remove from the heat.
In a food processor or blender, combine the cooked vegetables with the thickened stock and puree until smooth, about 2 minutes. Add the remaining 1-1/4 cups of the broth and puree 30 seconds more. Season to taste with salt, transfer to a bowl and refrigerate until chilled, about 1 hour.