Assemble, prepare, and measure ingredients. Crack eggs into teacups or ramekins. Thinly slice garlic. Line a plate with paper towels.
Bring a small saucepan of water to a boil. Turn down heat, make a whirlpool with a wooden spoon, and gently slide in two eggs. Set a timer for 3 minutes. When eggs are poached, carefully remove them to the plate with a slotted spoon. Repeat with the other two eggs.
In a skillet large enough to hold all the spinach, warm olive oil over medium heat and stir in garlic. Sauté garlic very briefly, then toss spinach in by the handful, adding more as it wilts. Once spinach has all wilted, turn off heat. Season with salt and pepper, remove to a bowl, and keep warm.
In the same skillet, add some olive oil if necessary and quickly crisp prosciutto slices, about 30 seconds per side.
To serve, make a nest of spinach on each of four plates. Top with a poached egg, season with salt and pepper, and serve with a couple of prosciutto slices on the side.