In a pot, bring vegetable stock to a simmer; keep warm.

Heat a large saucepan over medium-high heat. Add 3 teaspoons oil and rice to the hot pan and toast for 3 minutes, stirring occasionally. Add 2 1/2 cups hot stock, salt and pepper. Bring to a rapid boil and then reduce to a simmer. Cover and cook for exactly 18 minutes. Turn off the heat and spread the rice on a baking sheet lined with parchment paper. Let cool slightly while preparing remaining ingredients.

Preheat oven to 425° F.

Toss mushrooms with 1 tablespoon oil, salt and pepper. Roast in the oven until just golden, about 10 minutes.

Heat remaining oil in a large saucepan over medium-high heat. Add onions and cook until soft, about 2 minutes. Stir in roasted mushrooms and cook 1 minute more. Deglaze the pan with wine and let reduce for 1 minute. Stir in remaining 1/4 cup vegetable stock and heavy cream; bring to a simmer. Gently fold in cooled risotto and cook for 1 minute. Remove from heat, stir in cheese and allow to cool slightly.

Divide risotto into 4 balls and gently flatten using parchment paper. Lightly spray each cake with oil and roast on a parchment-lined baking sheet until just browned, 7 to 10 minutes.