Buying precooked beets, which I see at more and more grocery stores these days, makes this soup incredibly quick and easy to throw together, but you can use your own home-cooked beets if you like by baking or steaming beets until tender, then peeling off the skins. If making the bread crumbs ahead of time, store the cooled crumbs in an airtight container for up to three days, then reheat in a skillet before stirring in the lemon zest and feta to serve.
Prepare your charcoal or gas grill for direct, high-heat grilling. (Or heat your broiler to high and line a baking sheet with foil.) Brush the leek halves and cabbage wedge all over with olive oil, season with salt and pepper, and grill or broil the leeks and cabbage, flipping once halfway through, until deep golden brown all over with some charred bits at the edges, about eight minutes. Remove the leeks and cabbage from the grill or broiler, let cool then roughly chop.
Place the leeks and cabbage in a blender along with one-half cup olive oil, the beets, water, garlic and lemon juice. Blend until smooth, at least 30 seconds, then pour into a large bowl and refrigerate until chilled, at least one hour.
While the soup chills, make the spicy feta bread crumbs: Place the bread in a food processor and pulse until it forms rough crumbs, some fine and some the size of peas. You should get about 1 1/2 cups. Place a quadruple-thick layer of paper towels on a plate and keep nearby. Heat the remaining one-half cup olive oil in a large skillet over medium-high heat until the oil begins to shimmer. Add the crumbs (they will look sodden in the oil; it’s OK) and cook, stirring, until golden brown and crisp, six to seven minutes. Stir in the chile flakes and cook for 30 seconds. Immediately scrape the crumbs onto the paper-towel-lined plate and spread out in an even layer to cool. Once cooled, transfer the crumbs to a bowl and stir in the reserved lemon zest. Season with salt and pepper. (See note above if making ahead of time.)
Remove the soup from the fridge and season with more salt and lemon juice. Stir the feta into the bread crumbs right before you’re ready to serve them. Pour the soup into bowls and scatter the spicy feta bread crumbs over the top.
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