How to Make It
Preheat the oven to 375°. Arrange the garlic halves cut side up on a sheet of foil, drizzle with olive oil and wrap tightly. Roast until the garlic is very tender, about 1 hour. Let cool, then squeeze the cloves out of their skins into a small bowl and mash to a paste.
Bring a saucepan of salted water to a boil. Put the asparagus in a strainer, submerge it in the water and cook just until crisp-tender, 2 minutes. Cool the asparagus under running water and pat dry.
Return the water to a boil. Add the fregola and cook until very tender, about 20 minutes. Drain well and let cool slightly.
In a food processor, puree the garlic paste, fregola, chickpeas and the 2 tablespoons of olive oil; transfer the mixture to a bowl. Stir in the asparagus, artichokes, olives, scallions, yogurt, lemon juice, herbs and quinoa. Season with salt and pepper.
Heat a large cast-iron griddle or skillet and brush with olive oil. Using lightly moistened hands, form the mixture into eight 3-inch patties. Brush the patties with oil and cook over high heat until golden on the bottom, 4 minutes. Flip the patties and top each with a slice of cheese. Reduce the heat to moderate, cover and cook until the patties are heated through and golden on the bottom and the cheese is melted. Set the burgers on the buns and top with sun-dried-tomato spread, marinated mushrooms and sliced tomatoes.