Creamy Polenta with Balsamic Mushrooms and Onions

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Polenta:

4 cups (1 liter) low-sodium vegetable or mushroom broth

2 garlic cloves, finely grated

1 cup (150 grams) yellow cornmeal

2 tablespoons (28 grams) butter

1/2 cup (50 grams) grated Pecorino cheese

2 tablespoons finely chopped parsley

Kosher salt

Freshly ground black pepper

Mushrooms and Onions:

4 cups (about 450 grams) mixed mushrooms, roughly torn, sliced or halved

Kosher salt

2 tablespoons (28 grams) butter

1 red onion, cut into thin wedges

1 sprig rosemary

Freshly ground black pepper

2 garlic cloves, thinly sliced

1/4 cup (60 milliliters) balsamic vinegar

2 teaspoons flour

1 1/2 cups (375 milliliters) low-sodium vegetable or mushroom broth

1/2 lemon, zested and juiced

2 tablespoons finely chopped fine herbs, such as parsley, tarragon, basil or chives, plus more for serving

Shaved Pecorino, for serving