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1
Preheat an oven to 400 degrees F (200 degrees C).
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2
Cut the tops off the beets and place them in a baking dish with enough water to cover the bottom of the dish.
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3
Roast the beets in the preheated oven until they are easily pierced with a knife, 40 to 50 minutes. Refrigerate until cool. Slice and set aside.
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4
Preheat the oven's broiler and set the oven rack to the second level from the heat source.
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5
Whisk the olive oil, balsamic vinegar, rosemary, and 2 cloves minced garlic together in a bowl. Spread about half of this mixture over the ribbed side of the portobello mushroom caps; arrange the mushrooms on a baking sheet with the ribbed sides facing upwards.
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6
Broil the mushrooms until tender, making sure to not burn the garlic, 5 to 7 minutes. Flip the mushrooms and brush the remaining olive oil mixture over the tops of the caps. Return to the oven and broil until tender, about 5 minutes more.
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7
Spread equal amounts of the goat cheese on one half of each of the sandwich rolls. Top each with a portion of the sliced beets and the spinach. Whisk the mayonnaise, garlic, and lime juice together in a bowl; spread evenly over the remaining sandwich roll halves and top with one mushroom cap each. Bring the two halves together to form the sandwiches to serve.
