Prepare the beets, if making your own, a day before you want to make your sandwiches. (If using store-bought pickled beets, skip to Step 5.) Heat the oven to 400 degrees.

Put the beets in a small roasting pan. Drizzle with the olive oil and season with salt. Add just enough water to the pan to evenly cover the bottom by about 1 inch. Cover with aluminum foil and roast until tender when pierced with a knife, about 1 hour. Let the beets cool until you can handle them, then peel them, slipping the skins off with your fingers or a kitchen towel and using a paring knife where they stick. Cut into ¼-inch slices and place in a large, heatproof bowl.

In a saucepan, combine 1 cup water with the remaining pickled beet ingredients and ½ tablespoon salt. Bring to a boil over medium-high heat, stirring to dissolve the sugar and salt. When the sugar and salt have dissolved, pour the pickle brine over the sliced beets.

Let the beet pickles cool at room temperature and then put the beets and their brine into a plastic or glass container, cover and refrigerate. The pickled beets will be ready to eat the next day and will keep for up to 2 months.

To make the sandwiches, toss the carrots and apple cider vinegar together in a small bowl; set aside 15 minutes.

In another small bowl, stir together the mayonnaise, pimentón and garlic.

Split the focaccia squares in half horizontally and arrange on a large cutting board or baking sheet, cut sides up. Spread both cut sides with the pimentón mayonnaise. Arrange the egg slices evenly on the bottom half of the bread.

Drain the carrots and add them to a large bowl, along with the olives, capers, herbs and sliced beets. Add just enough olive oil to coat lightly and toss until combined. Mound the salad on top of the egg. Slice the feta and arrange on top of the salad (or crumble it on top).

Quickly replace the top of the bread before the sandwich falls apart, pressing gently to help it hold together, and serve right away.