Heat 2 teaspoons of the vegetable oil in a heavy saute pan over high heat. Season both sides of the halibut fillets with salt and pepper.
Once oil is hot, add seasoned halibut. Sear the fish until it's just cooked through, 3 to 4 minutes each side, depending on thickness. Remove to a platter and keep warm.
Add remaining oil to the saute pan and, over medium-high heat, saute onions, garlic, fava beans and shiitake mushrooms until the onions are tender, about 3 to 4 minutes.
Add chicken broth, bring to a boil and cook until reduced by half, scraping up any browned bits from the bottom of the pan.
Remove from heat and whisk in butter, tarragon and lemon juice, and season to taste with salt and pepper.
To serve, place rice in middle of each plate and place halibut fillet on top of rice. Using a slotted spoon, spoon the fava bean and shiitake mixture over the fillets, then drizzle with the pan sauce. Garnish with reserved green onion tops.
Note: Find Walla Walla spring onions at farmers markets and specialty markets. You can substitute 4 to 6 scallions.
Note: To blanch and peel fava beans, boil shelled beans for about 50 seconds. Remove beans from pot and immediately plunge into cold water. Drain beans and remove whitish skins, using your fingernail to open the spongy skins, and then pinch the beans free.