In a large mixing bowl, whisk together 3 cups Greek yogurt and 2 tablespoons olive oil until smooth.
Add 1 cup of the low-fat milk and whisk to loosen the yogurt; continue adding additional milk a little at a time until the mixture reaches a soupy consistency you prefer (about 1–2 cups total).
Grate the Persian cucumbers on a box grater into a bowl.
Wrap the grated cucumbers in cheesecloth or a clean kitchen towel and squeeze firmly to remove excess moisture.
Stir the drained cucumbers into the yogurt mixture along with 1/3 cup chopped dill, 1 teaspoon crushed garlic, 1/2 teaspoon salt and white pepper to taste; adjust seasoning as needed.
Cover and chill in the refrigerator for at least 1 hour and up to 4 hours to allow flavors to meld.
Before serving, whisk the soup again and add a splash more milk if needed to loosen the texture; taste and adjust seasoning or lemon juice if desired.
Ladle into bowls and garnish with fresh mint and optional chopped walnuts; serve chilled.