Preheat oven to 350°. Whisk together eggs, cream, and 1/4 tsp. salt.
Cook potatoes in hot oil in a 10-inch ovenproof nonstick skillet over medium-high heat, stirring often, 1 minute. Sprinkle with remaining 1/4 tsp salt. Reduce heat to medium, and cook, stirring often, 3 to 4 minutes or until light golden brown on both sides. Transfer potatoes to a bowl.
Return skillet to heat; melt 1 tsp. butter in skillet. Add zucchini; cook, stirring often, 2 to 3 minutes or until crisp-tender. Add remaining 2 tsp. butter to skillet; let melt. Add potatoes, stirring to coat.
Pour egg mixture over potatoes. Reduce heat to medium-low; cook 1 minute. Sprinkle with cheese.
Bake at 350° for 8 to 9 minutes or until center is set. Remove from oven; invert onto a serving plate.