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1
Heat the vegetable oil to 375F in a deep fryer or a deep, heavy pan.
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2
In a bowl, season the flour with the salt and pepper.
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3
In another bowl, whisk 2 of the eggs with water to make an egg wash.
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4
Spread the panko on a plate.
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5
Dip the tomato slices, one at a time, into the seasoned flour, transfer to the egg wash, and then toss in the panko until coated.
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6
Carefully place them in the hot oil and fry until golden brown.
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7
Using a slotted spoon, transfer to a paper towellined plate to drain.
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8
Heat a large griddle over medium heat, brush the buns with 1 tablespoon of the melted butter, and toast on the griddle until golden brown.
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9
Set aside.
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10
Form the beef into 4 patties of equal size.
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11
Liberally season with salt and pepper.
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12
Heat a large griddle pan or stovetop grill pan over medium-high heat and add the burgers to the pan.
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13
Cook, turning once, until medium-rare, 2 to 3 minutes, or longer if desired.
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14
While the burgers are cooking, heat a large nonstick pan over high heat, add the remaining 2 tablespoons butter, and crack the remaining 4 eggs into the pan, frying until cooked over easy, or over medium if preferred.
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15
To assemble, spread 1 ounce of the pimento cheese on the top half of each bun, then layer 1 burger, 1 egg, and 2 slices of the fried green tomatoes on the bottom of the bun.
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16
Assemble the two halves and serve immediately.