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For SAUCE:
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Heat oil in heavy large skillet over medium heat.
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Add shallots and saute 15 minutes.
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Sprinkle with sugar and stir until shallots are golden brown, about 5 minutes.
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Add vinegar and boil until reduced to syrup consistency, about 1 minute.
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Add both broths and boil until liquid is reduced to sauce consistency, about 10 minutes.
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Add thyme.
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Season with salt and pepper. (Can be made 1 day ahead. Cover and chill).
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Prepare barbecue (medium-high heat) or preheat broiler.
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For ONION RINGS:
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Preheat oven to 240u00b0F
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Pour vegetable oil into heavy large saucepan to depth of 3 inches.
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Heat to 375u00b0F
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Stir flour, salt and pepper in large bowl.
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Add onion slices and toss to coat, separating rings.
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Working in batches, shake excess flour off onions, add to oil and fry until golden, about 2 minutes per batch.
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Using slotted spoon, transfer onion rings to paper towels and drain.
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Place onions on large baking sheet.
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Season lightly with additional salt.
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Place in oven to keep warm.
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Season steaks with salt and pepper.
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Grill or broil steaks to desired doneness, about 4 minutes per side for rare.
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Transfer steaks to plates.
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Rewarm sauce.
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Spoon shallot sauce around steaks.
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Top steaks with onions.