Bring water to a boil. Add quinoa; cover, reduce heat and simmer for 15 minutes. Remove from heat and let stand, covered, until quinoa is tender and liquid is absorbed, about 5 minutes. Fluff with a fork. Preheat oven to 350°F; line a large baking sheet with foil. Heat a medium skillet over medium heat. Add oil to pan; swirl to coat. Add shallots and garlic; cook, stirring occasionally, until tender, about 5 minutes. Gradually add spinach, tossing constantly until spinach wilts. Remove from heat. Stir in quinoa, cheese, dill, salt, and pepper. Spoon mixture evenly into mushroom cups (about 1 1/2 teaspoons each). Arrange mushrooms on a baking sheet. Bake until tender, about 25 minutes.