Place the dried morels in a bowl and cover with boiling water. Allow the mushrooms to rehydrate for 20-30 minutes. Remove the mushrooms from the water, reserving the liquid, and slice in half or quarters for larger mushrooms.
Strain the reserved mushroom liquid through cheesecloth or a fine mesh strainer into a large liquid measuring cup. Adjust, adding more water if necessary, to reach a volume of 1.5 cups.
Pour the mushroom liquid into a medium saucepan and bring it to a boil. Stir in the quinoa along with a pinch of sea salt. Return to a boil, cover the pot, then reduce the heat to low. Simmer for 15 minutes, or until the quinoa has absorbed all the liquid, then remove the pot from heat.
Meanwhile, heat a large skillet over medium-low heat. Add the pine nuts and toast until fragrant and golden brown, about 5-6 minutes. Transfer the pine nuts to a dish and set aside.
Return the skillet to medium heat and add two tablespoons of olive oil. Add the shallot and cook until soft, about 2-3 minutes.
Add the sliced morels to the skillet and sauté until they are fragrant and slightly browned, about 8-10 minutes.
Add the garlic, thyme and small pinch of sea salt to the skillet and cook until the garlic is fragrant, about 1-2 minutes.
Add the arugula to the skillet and cook until it is just wilted, 1-2 minutes, then remove the skillet from heat.
Combine the quinoa and mushroom mixture together in a large bowl. Add the remaining tablespoon of olive oil, lemon juice, black pepper, and toss. Top with the toasted pine nuts and crumbled cheese.