Season the halibut fillets with thyme leaves and grated lemon zest. Cover and refrigerate for 4 hours.
15 minutes before cooking the halibut, remove from refrigerator.
Heat a large sauté pan over high heat for a few minutes. Season the halibut with Kosher sea salt and freshly ground pepper. Add 1 and 1/2 tablespoons extra virgin olive oil to the pan and warm for a minute. Add the halibut fillets to the pan and cook for 3-4 minutes or until lightly golden. Turn the fillets, lower the heat to medium-low, and cook for two more minutes or until cooked through.
To serve; spoon 1/4 of the spring vegetables on the center of a plate, top with one of the halibut fillets and drizzle a teaspoon of the parsley-mint sauce over the halibut.
Parboil the carrots and potatoes for one minute in salted water. Drain well.
Heat a large sauté pan over medium high heat, add the olive oil and 1/2 tablespoon butter. When the butter has melted, add the carrots and potatoes sauté for 2-3 minutes. Continue to add remaining vegetables stirring and tossing as you add them. Season with salt and pepper. Sauté until vegetables are just tender. Add the remaining 1/2 tablespoon of butter and a squeeze of lemon.
In a small mini food processor, add the parsley, mint, garlic, lemon juice, salt, pepper, and half the olive oil. Pulse until combined. Add the remaining olive oil and pulse to mix. Taste and adjust for seasoning.