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Beignets: In a medium pot, add the first 5 ingredients and bring to a boil over medium heat.
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At this point, add in the flour and stir vigorously with a spoon.
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Continue to cook until the dough becomes a solid mass and begins to pull away from the sides of the pot, about 3 to 5 minutes.
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Switch off the heat, mix in the vanilla extract and add in the eggs, 1 at a time, incorporating each egg before adding in the next.
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5
(Chef's Note: Work quickly or else the eggs will curdle.)
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The resulting pastry should be flexible and soft, but firm enough to hold its shape.
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Set it aside covered with plastic for 5 minutes.
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When ready to make the pastries, fill a deep skillet or deep-fryer 2/3 full of oil and heat it to 300 to 325 degrees F. Form the dough into doughnut hole shapes.
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Fry the beignets, in batches, until golden brown, about 3 to 6 minutes being sure not crowd them.
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Remove them from the oil to a cooling rack.
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Caramel: Put the rosemary and water into a small pot and simmer over medium heat until the rosemary is limp, about 3 to 6 minutes.
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Strain the liquid, discarding the rosemary, and pour it back into the pot.
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(Chef's Note: If this liquid is not 1/4 cup exactly, add water.)
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Add in the sugar and corn syrup.
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Bring the ingredients to a boil over high heat, then reduce the heat to medium and simmer until the syrup thickens.
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Stir in the heavy cream and butter and switch off the heat.
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Pour into a serving bowl and reserve for the beignets.
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Arrange the beignets on a serving platter and dust with powdered sugar.
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Serve with the honey and Rosemary Caramel in small bowls.
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Food Network Kitchens have not tested this recipe and therefore, we cannot make representation as to the results.