For the soup: In a medium-sized pot, heat oil over medium-high heat. When the oil is hot, add the minced garlic and cook for 1 minute. Add 6 cups of water to the pot. Add the potato and the edamame and cover the pot with a lid. Bring the water to a boil, then remove the lid and simmer until the vegetables are tender (15 to 20 minutes). Season with salt and pepper. Blend the soup and cover it to keep it warm.
For the topping: In a medium skillet, heat oil over medium-high heat. Add the small cubes of black forest ham and shiitake mushrooms. Cook for about 5 minutes and then add the spinach. Remove from the heat. Season with salt and pepper.
To serve: Reheat the soup a little bit and pour into 4 bowls. Use a spoon to add the mushrooms, ham, and spinach mix.