In a large nonstick skillet with a lid, heat 1 teaspoon oil over medium-high. Add mushrooms, and season with salt and pepper. Cook, stirring occasionally, until mushrooms are browned and have released their liquid, 6 to 8 minutes. Transfer to a bowl; set aside.
Reduce heat to medium-low. In same skillet, heat 1 teaspoon oil. Add onion and cook, stirring occasionally, until translucent, 3 to 5 minutes. Stir in couscous, 1 1/4 cups water, and 1/2 teaspoon coarse salt; bring to a boil. Reduce to a simmer, cover, and cook until couscous is tender, 8 to 10 minutes (liquid should be completely absorbed). Stir in dried cranberries.
Meanwhile, make vinaigrette: In a large bowl, whisk together remaining 4 teaspoons oil, lime juice, and mustard; season with salt and pepper. Pour half the vinaigrette into another large bowl. To first bowl, add couscous mixture and mushrooms (along with any accumulated juice), and to the other, spinach. Toss contents of each bowl until well combined.
Divide spinach among 4 serving plates; top with turkey slices, and serve with couscous.