Trim the tails from the asparagus spears yielding 10 tips of the same length and about 200 grams of stems. Peel the tips and set aside.
In a small sauce pan, bring some water to a boil and cook the stalks until well cooked, but still green. Drain. Purée stems in a food processor. Strain purée through a fine strainer. Set aside. There should be about 100 milliliters of asparagus purée.
Place the chicken stock (200 milliliters) in a small sauce pan and reduce substantially. Add cream and reduce further. Set aside.
Place the remaining chicken stock in a shallow pan large enough to hold the chicken breasts in a single layer. Bring to a boil. Add breasts and reduce to a simmer. Poach until almost cooked, about 7 minutes. Drain, set aside, and tent with foil for 5 minutes.
Meanwhile, bring some salted water to a boil in a sauce pan. Cook asparagus tips until tender, but still bright green. Drain well on paper towels. Start cooking the asparagus tips just when the chicken is removed from its poaching liquid.