Preheat oven to 425 degrees. Line a large baking sheet with parchment.
Heat the broth in a small saucepan over medium-low heat until steaming. Do not boil.
Smash 4 of the garlic cloves and toss with 2 tablespoons olive oil, the chickpeas, mushrooms, 1 tablespoon of oregano, salt, and pepper. Spread onto the baking sheet.
Roast for 10 minutes. Arrange 2 tablespoons of thinly-sliced butter over the mushrooms and chickpeas. Increase the oven temperature to 450 and roast an additional 5-10 minutes until crispy and golden. Set aside.
Melt remaining 2 tablespoons of butter in a large pot over medium heat. Mince the other 4 cloves of garlic and stir with 1 tablespoon of oregano for 1-2 minutes until fragrant. Pour in the rice and toast for 3-4 minutes, then add the white wine and simmer until absorbed by the rice, about 3-4 minutes longer.
Pour in 3 cups of the hot broth and simmer, covered, for 15 minutes until the rice is softened. Stir in 1/2 cup of broth, the parmesan, and pesto. If needed, add the remaining broth to loosen the sauce.
Divide into bowls and garnish with the chickpeas and mushrooms along with fresh chopped basil, as desired.