Soak dried porcini mushrooms in hot water for 15 minutes, until soft.
Cook penne according to package directions.
While penne is cooking, heat oil in large skillet over medium-high heat. Add leeks and cook for 2 minutes, or until soft. Add shiitake mushrooms and cook for 3 minutes, until mushrooms soften. Strain porcini mushrooms through a sieve, reserving soaking liquid. Chop porcini mushrooms and add to skillet. Add chicken and tarragon and cook 1 minute, until tarragon is fragrant.
Add reserved mushroom soaking liquid and 1 cup of the chicken broth and bring to a simmer, for 5 minutes. In a small bowl, using a whisk, dissolve cornstarch in remaining 1/4 cup chicken broth and add to the pan with the peas. Simmer 2 minutes until it reaches a full boil, and the sauce thickens. Stir in cooked penne and cook 1 minute to heat through. Remove from heat and stir in parsley. Season, to taste, with salt and freshly ground black pepper.
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