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Preheat a large cast-iron pan over medium-high heat for 5 minutes.
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In a small bowl, whisk together 2 tablespoons of the olive oil and the capers, lemon juice and zest, and chopped herbs.
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In a large bowl, toss the broccolini with 1 tablespoon of the remaining olive oil and the salt, cumin, and chili flakes.
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Working in batches, if necessary, place the broccolini in the hot pan in one single layer. Cook for 2 minutes, untouched, until slightly charred, then, using tongs, flip broccolini over. Cook for another 2 minutes; remove onto a plate.
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Add the remaining 1 tablespoon olive oil to the pan and cook the eggs until the whites are crispy and yolks runny. Season the yolks with a bit of salt.
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Warm up the tortillas if needed, then place two each on four dinner plates. Top each with a fried egg, a few charred broccolini, and the caper dressing.