Place eggs in a saucepan and cover with cold water about one inch above the eggs. Bring to a full boil on high heat, then turn off the heat. Cover the pot and let sit for 12 minutes. Peel eggs under cold running water.
Chop the peeled eggs into small, rough pieces. Do not chop too fine or the egg salad will become mushy.
In a bowl, combine chopped eggs, mayonnaise, Dijon mustard, and sea salt. Stir gently until all egg pieces are evenly coated. Taste and adjust seasoning if needed.
Using a round cookie cutter, cut 2 circles from each slice of bread, avoiding the crust. You will need 12 bread circles total for 6 sandwiches.
Spoon egg salad onto one bread circle. Place a second bread circle on top. Skewer a small sprig of flat-leaf parsley with a cocktail fork and press gently into the top of each sandwich.
Let sandwiches sit at room temperature for 15–20 minutes before serving. Do not serve cold.