Heat the olive oil in a large pot over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes. Stir in the minced garlic and cook for another minute until fragrant.

Add the diced potatoes to the pot, stirring them thoroughly to ensure they are well-coated with the onion mixture. Cook over medium heat for 2-3 minutes, allowing the potatoes to soften slightly and release their starches, which will help thicken the soup later.

Pour in the chicken broth, ensuring the potatoes are fully submerged. Bring to a gentle simmer, then reduce the heat to low. Cover and cook for 15-20 minutes until the potatoes are tender.

Remove the pot from heat and use a stick blender to blend the soup until smooth, leaving some potato chunks for texture. Alternatively, blend half the soup in a blender and return to the pot.

Gently stir in the milk and salt, ensuring they are well combined. Place the pot back on low heat, allowing it to simmer gently for 5 minutes. Stir occasionally to prevent sticking. Taste the soup and adjust the seasoning if necessary, ensuring a balanced flavor.

To achieve a thicker consistency, use a fork to mash some of the cooked potatoes directly in the pot. This action releases additional starch, which contributes to the soup's creamy texture. Ensure you leave some potato chunks for added texture before proceeding to blend the mixture until smooth.

When reheating the soup, gently warm it over low heat, stirring occasionally to prevent sticking. If it appears too thick, gradually incorporate a splash of milk, stirring until the soup reaches your preferred consistency. Avoid boiling to maintain the soup's creamy texture and prevent curdling.

If the soup seems bland, adjust the seasoning by adding a small pinch of salt. Stir well and taste again, ensuring the flavors are balanced. Be cautious not to over-salt, as the soup should have a gentle, savory taste. Repeat if necessary, tasting after each addition.

Once the soup is ladled into bowls, sprinkle each serving with chopped fresh parsley. This adds a vibrant green touch and a hint of herbal aroma. The parsley should be added just before serving to maintain its bright color and fresh taste, enhancing the soup's presentation.

Ladle the hot soup into bowls, ensuring each serving has a generous portion of potatoes and broth. Garnish with a sprinkle of chopped fresh parsley for a touch of color and aroma. Serve immediately with crusty bread or a fresh side salad to complement the soup's creamy texture and warmth.