In a 12-inch skillet, melt 2 Tbs. of the butter over medium heat. Add the mushrooms and sauté until softened and browned, 6 to 8 minutes. If flavorful browned bits are sticking to the bottom of the pan, stir in 1 or 2 Tbs. of water to deglaze the pan. The water will evaporate quickly. In a large bowl, toss the hot mushrooms with the sliced spinach until well combined. The heat of the mushrooms will wilt the spinach. Stir in the chives and 1/4 tsp. of the salt. If preparing a day ahead, cover and refrigerate until ready to use.
Position a rack 6 inches from the broiler element and heat the broiler on high. In a large bowl, whisk the eggs, cream, 2 Tbs. water, the remaining 1/2 tsp. salt, and 1/4 tsp. of the pepper. Stir the mushroom and spinach mixture into the eggs.
In a 12-inch ovenproof skillet, heat the remaining 2 Tbs. butter over medium-high heat, swirling the pan so the butter coats the sides. Add the eggs to the pan, evenly distributing the spinach and mushrooms, and sprinkle with the remaining 1/4 tsp. pepper. Cook until the eggs begin to set and the bottom begins to brown, occasionally lifting the sides with a spatula to let uncooked egg run underneath, 3 to 5 minutes. Transfer the pan to the broiler. Broil until the eggs are almost fully set and the top puffs and browns, 1 to 5 minutes, depending on your broiler. Check the frittata every minute to avoid overbrowning. Let stand for 5 minutes before sliding out of the pan onto a cutting board. Slice into squares or wedges and serve warm or at room temperature.