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Preheat the oven to 400F/220C.
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In a mixing bowl, toss the broccoli florets with 3 tablespoons of the olive oil, garlic, salt and pepper. Spread onto a sheet pan and roast for 8 to 10 minutes, or until the broccoli is lightly charred and softened. Remove from the oven, toss with dried oregano and red-pepper flakes, and allow to cool.
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Heat a skillet over medium-low heat.
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Brush one side of each slice of bread with remaining olive oil. Dividing the ingredients evenly, layer cheese on the unoiled side of half of the bread slices, then add chopped, cooked broccoli in the middle and more feta on top so that the broccoli lies between two layers of cheese.
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Top each with another slice of bread,, oiled side out.
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Heat a nonstick or cast-iron skillet over medium heat and drizzle in a teaspoon of olive oil, tilting the skillet to distribute it evenly across the surface. Place the first broccoli-feta sandwich in the skillet and press down with a spatula. Sear until the bread is golden brown on the bottom side, then turn over once, and continue “grilling” on the other side. Remove and repeat with remaining sandwiches, replenishing the olive oil in the skillet as needed. Keep the grilled cheese sandwiches tented with aluminum foil to keep warm until they’re all done. Serve hot!