Place the eggs in a small pot and cover with cold water by about an inch. Bring to a simmer over medium-high heat, and simmer for three minutes. Turn the heat off, and cover the pot for an additional 10 minutes. Drain the eggs and immediately cool them down with cold water. Peel.

Remove the yolk of one of the hard-boiled eggs and mash in a medium sized bowl. Whisk the vinegar into the yolk, followed by the mustard, whisking each ingredient into the dressing for uniform consistency.

Slowly add ½ cup of olive oil beating constantly with a wire whisk; the dressing should look smooth and glossy. Stir in the capers, shallots, herbs, and salt.

Press the three remaining hard-boiled eggs through a sieve. I used a potato ricer with the smallest holes, and it worked beautifully. Do the same with the leftover egg white from the fourth egg. Stir into the dressing using a fork.

Preheat the oven to 400˚F and line two baking sheets with parchment paper.

Cut the potatoes into 1-inch chunks and toss with 1 tablespoon olive oil. Turn onto a baking sheet, and lightly sprinkle with salt. Toss the broccoli florets in 1 tablespoon oil and add them to the second baking sheet; lightly sprinkle with salt.

Slide both baking sheets into the oven and roast until cooked through and browned (about 30 minutes); do not stir or move the potatoes or broccoli florets during this time for best results. Both should show signs of browning with a slight char on the broccoli.

Place the warm potatoes and broccoli into a large bowl and gently fold in three-quarters of the dressing.

Taste and adjust seasoning if needed, adding more of the dressing if desired.