Step 1Trim the tough ends from the asparagus, reserving the ends for flavoring the broth. Cut the asparagus spears in 1-inch pieces. Cut off 1 thin slice from the tip of each mushroom stem, and reserve the slices to flavor the broth.
Step 2To make the broth, combine the asparagus ends, mushroom stem slices, tarragon and parsley stems in a saucepan. Add the water, cover and bring to a boil. Cover and cook over low heat until the broth is aromatic and delicately flavored and the asparagus ends are tender when pierced with a knife, about 30 minutes. Strain and discard the vegetable and herb trimmings.
Step 3Return the broth to a boil. Add the asparagus pieces and a pinch of salt. Return to a boil and cook uncovered over medium-high heat until the asparagus is crisp-tender, 3 to 4 minutes. Remove the asparagus with a slotted spoon, rinse briefly with cold water and drain. Pour the broth into a measuring cup and add water if necessary to make 2 cups. Dry the pan.
Step 4Heat 1 tablespoon oil and 1 tablespoon butter in the saucepan over medium heat. Add two-thirds cup chopped onion and cook, stirring occasionally, until softened, about 4 minutes. Add the garlic and cook, stirring, for 30 seconds. Add the bulgur and cook, stirring, for 2 minutes. Add the hot asparagus broth, standing back to avoid any steam. Add one-half teaspoon salt and a pinch of pepper. Stir and bring to a boil. Reduce the heat to low, cover and cook without stirring until the broth is absorbed and the bulgur is tender, about 15 minutes. Add the red pepper and mix in lightly. Cover the bulgur and set aside for 5 to 10 minutes.
Step 5Meanwhile, halve the mushrooms and slice them about one-fourth-inch thick.
Step 6Toast the pine nuts in a skillet over medium-low heat, shaking the pan often, until lightly brown, 2 to 3 minutes. Set aside to cool.
Step 7Heat the remaining oil and butter in a large skillet over medium heat. Add the remaining onion and cook, stirring occasionally, until it begins to turn golden, about 5 minutes. Add the mushrooms, salt and pepper and saute over medium-high heat until they are tender, about 3 minutes. If the mushrooms give off liquid, cook the mixture over high heat until it evaporates. Add the asparagus, salt and pepper to taste and saute over medium heat until it is heated through, 1 to 2 minutes.
Step 8Fluff the bulgur with a fork and stir in the chopped tarragon and parsley. Stir in the asparagus-mushroom mixture (set aside a few asparagus tips for garnish). Taste and add additional seasoning, tarragon or red pepper if needed. Drizzle over a little olive oil if desired. Serve the pilaf garnished with chopped tarragon and parsley, asparagus tips and toasted pine nuts. This makes about 5 cups pilaf.