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1
In a medium saucepot over medium heat, melt 2 tablespoons of the butter.
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2
Add the cognac, onions and thyme and cook until the onions are soft and caramelized.
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3
Add the stock and continue cooking, stirring occasionally, until most of the liquid has evaporated, about 20 minutes total.
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4
Remove the thyme sprigs and set the mixture aside.
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5
In a heavy-bottomed saucepan, melt the remaining 3 tablespoons butter.
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6
Using a whisk, whisk in the flour and cook, stirring constantly, until the flour does not have a raw flavor, about 2 minutes.
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7
The mixture will bubble a bit; do not let it brown.
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8
Whisk in the milk, making sure the mixture is smooth and there are no lumps.
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9
Continue to cook and stir as the sauce thickens.
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10
Whisk in 8 ounces of the cheese until melted and bring the mixture to a boil.
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11
Add salt and white pepper to taste, lower the heat and cook, stirring, for 2 to 3 minutes more.
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12
Remove from the heat and set aside.
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13
(To cool this sauce for later use, cover it with plastic wrap to prevent a skin from forming on the top of the sauce.)
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14
Pile up the remaining 12 ounces grated cheese on some wax paper and mold the cheese back together with your hands.
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15
Roll the cheese out as if you were rolling dough.
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16
With a 4-inch ring, cut out 6 rounds.
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17
Set aside.
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18
Using the same ring, mold out 6 burgers with the ground beef, each around 3/4 inch high.
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19
Heat a grill pan over high heat.
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20
Lightly brush the pan with olive oil.
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21
Brush both sides of the burgers with olive oil.
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22
Sprinkle on a generous amount of salt and some brown sugar.
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23
Cook the burgers on the grill pan for 5 to 6 minutes on the first side.
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24
Flip the burgers, top with the cheese and cook until the cheese melts and the burgers are cooked to desired doneness, 5 to 6 minutes for medium.
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25
Butter the toasted English muffins.
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26
Spread 1 teaspoon of the Dijon mustard on the bottom halves of each muffin.
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27
Place the burgers on top and then top with a tablespoon of the sauteed onions, followed by a tablespoon of the bechamel sauce.
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28
Place the other halves of the muffins on top and serve.
