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1
Preheat the grill to medium-high.
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2
Melt the butter over medium-high heat in a large skillet.
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3
Add the onions and rosemary and cook until the onions are slightly translucent, about 2 minutes.
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4
Stir in the mushrooms and cook, stirring, until the mushrooms are soft and most of the juices have evaporated, 4 to 5 minutes.
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5
Pour in the wine and simmer until most of the liquid has evaporated, about 2 minutes.
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6
Season with 1/4 teaspoon each salt and pepper.
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7
Cut a deep pocket horizontally into each steak.
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8
Sprinkle the steaks with salt and pepper.
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9
Stuff each pocket with about 1 1/2 tablespoons of the mushroom mixture, reserving the remaining mushroom mixture for serving.
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10
Skewer with toothpicks to close.
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11
Brush the steaks with olive oil.
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12
Place the steaks in a grill basket (this way any filling that falls out during cooking won't fall into the fire).
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13
Transfer the basket to the grill.
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14
Close the cover and cook to desired doneness, 5 to 6 minutes per side for medium-rare.
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15
Let the steaks rest for 5 minutes on a platter or cutting board
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16
Serve the steaks with the remaining mushroom mixture and the Bearnaise Sauce.
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17
Combine the wine, half the tarragon, the vinegar, garlic, shallots and some salt and pepper in a saute pan over medium-low heat.
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18
Cook until reduced by half.
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19
Transfer to a medium metal bowl.
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20
Start to whisk the wine mixture in the bowl while adding the butter in increments.
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21
(Do not stop whisking throughout this whole process.)
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22
Temper the eggs so they do not scramble and add them to the bowl.
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23
The sauce will look glossy and thick.
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24
Whisk in the lemon juice, if using.
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25
Taste for seasoning.
