Heat oil in a large skillet over medium-high heat. Add zucchini and cook, stirring frequently, until soft, 4 to 6 minutes. Add shallots and garlic; cook, stirring frequently, until fragrant, about 30 seconds. Remove from heat. Stir in half the spinach; cover and let stand until wilted, 1 to 2 minutes. Stir in the remaining spinach; cover and let stand until wilted, about 5 minutes more. Uncover and allow the mixture to cool for 10 minutes.