Step 1

Whisk egg yolks in a small saucepan until smooth, about 1 minute. Add 1 tablespoon cold water, cornstarch, Dijon, 1 tablespoon lemon juice, and 1/2 teaspoon salt; whisk until thoroughly combined. Prepare a bowl of ice water large enough to dip bottom of saucepan into; set aside.

Step 2

Add 1 tablespoon cold butter to egg yolk mixture; place over very low heat. Cook, whisking constantly, until thickened and bubbles have disappeared, 1 to 2 minutes, being careful not to curdle egg mixture. (If egg mixture looks anything but perfectly smooth, remove from heat, and briefly plunge saucepan into ice water, whisking constantly.) When butter is completely incorporated and mixture has thickened, immediately add remaining 1 tablespoon cold butter; remove from heat, and whisk until smooth.

Step 3

While whisking egg yolk mixture constantly, slowly add 2 tablespoons melted butter, 1/4 teaspoon at a time, until mixture begins to thicken, about 1 minute. Whisking constantly, add remaining 10 tablespoons melted butter, 1 tablespoon at a time, until sauce is smooth, about 2 minutes. Whisk in cayenne, remaining 1/2 tablespoons lemon juice, and remaining 1/2 teaspoon salt. Add warm water, 1 teaspoon at a time, as needed until sauce is thin enough to spread out on its own a bit but still thick and billowy. Cover pan, and move to back of stove, or set over another pan with hot water in it, to keep sauce warm. Add more warm water to thin if needed after resting.

Step 4

Set a steamer basket in a medium saucepan, and fill pan with 1 inch of water. Bring to a simmer over medium; add cold eggs. Cover and cook eggs 7 minutes for runny yolks, increasing time by 30-second increments up to 8 minutes and 30 seconds for less runny yolks. Transfer eggs to ice bath; let stand 1 to 2 minutes. Carefully peel eggs.

Step 5

To assemble, drape each piece of sourdough toast with 2 prosciutto slices. Cut peeled eggs carefully in half lengthwise, and top each toast with 2 egg halves. Spoon 1/4 cup hollandaise over each toast; sprinkle with herbs and, if desired, black pepper.