1. Using a mortar and pestle (I used the food processor…lazy), pulverize the garlic and anchovies until a smooth paste.
2. Transfer the paste to a small saucepan. Add the butter and olive oil and bring to a simmer over low heat. Continue simmering for about 5 minutes. Remove the pan from the heat and stir in the lemon zest and lemon juice. Season with salt to taste. Note: This can be made ahead of time and stored in the refrigerator until ready to use. Just be sure to rewarm it slightly before serving — I used the microwave.

for the hard-boiled eggs:

1. Place the eggs in a saucepan and add enough cold water to cover. Bring the water to a boil and immediately turn off the heat. Let the eggs sit in the pan for 15 minutes. Remove the eggs to a bowl and allow to rest for at least 2 minutes. When the eggs are cool enough to handle, crack the eggs on each side and then roll them back and forth with your hand, breaking up the shell. Remove and discard the shell.

to assemble the sandwiches:

1. Arrange the arugula over each piece of bread. Cut the top 1/2 inch off the eggs and, using a spoon, scoop them out of their shells in large spoonfuls onto the bread. Note: I tried this, but failed. I just had an easier time peeling the eggs, gently mashing them with a fork, seasoning them with a pinch of salt and pepper and a few chopped scallions (just because I had them), and then spooning this mixture over the grilled bread. Drizzle each piece with a few spoonfuls of the warm bagna cauda. Serve more bagna cauda on the side just in case.