Prepare an ice bath and set aside.
Boil water and cook eggs for 7–7½ minutes.
Transfer eggs to the ice bath and chill 2 minutes.
Peel under running water and pat dry.
In skillet, melt butter over low heat.
Add garlic; cook 1 minute.
Stir in seasonings; cook 1 minute.
Add broth and simmer 6 minutes.
Stir in lemon juice and splash of water.
Taste sauce, add salt if needed.
Place eggs in bowl and pour sauce over.
Garnish with parsley and serve.
