Cut the tops off the beetroots and place them in a baking dish with enough water to cover the bottom of the dish.
Roast the beetroots in the preheated oven until they are easily pierced with a knife, 40 to 50 minutes. Refrigerate until cool. Slice and set aside.
Preheat the grill and set the rack to the second level from the heat source.
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 tray with the ribbed sides facing upwards.
Grill the mushrooms until tender, making sure not to 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 grill and cook until tender, about 5 minutes more.
Spread equal amounts of the goats cheese on one half of each of the hamburger rolls. Top each with a portion of the sliced beetroot and the spinach. Whisk the mayonnaise, remaining garlic and lime juice together in a bowl; spread evenly over the remaining hamburger roll halves and top with one mushroom cap each. Bring the two halves together to form the burgers to serve.