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Preheat oven to 450 degrees.
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Make the herb butter: Mix all the ingredients together until well blended. You will have leftover herb butter than can be refrigerated or frozen to use another time.
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Toss carrot slices with olive oil, salt and pepper and arrange on a parchment or non-stick foil lined sheet pan.
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Scatter the rosemary and thyme over the top and roast until tender, about 20 mins. Reduce oven temperature to 400 degrees after you remove the carrots.
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While carrots are roasting, combine the sugars and water and stir over high heat until sugar is dissolved.
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Cook the sugar until it reaches a deep amber, about 8 minutes. Be careful not to burn.
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Remove from heat and stir in the apple cider vinegar.
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Spray the mini muffin trays with non-stick spray and pour one tsp of caramel sauce into the bottom of each cup.
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Layer about 4 pieces of roasted carrot into each cup and sprinkle with a 1/2 tsp of parmesan and a 1/4 tsp of the herb butter.
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Cut out 32 rounds of puff pastry with a 1 1/2 inch (fluted if you have it) cookie cutter.
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Place the pastry rounds on the top of the carrots and press down slightly.
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Bake at 400 degrees until puffed and golden, about 15-20 minutes.
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Remove the pan, allow to cool for 2-3 minutes, then place another sheet pan over the top and flip, giving the tray a good knock.
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Remove the muffin tray and replace any carrots that stuck to the pan.
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The tartlets can be served warm or room temperature.