In a saucepan, simmer stock. In another saucepan, stir onion and 3 tbsp. butter over medium, until onion softens, 3 minutes. Add the rice and garlic and stir 2 minutes. Reduce heat to low, stir 1 cup stock into the rice and cook, stirring, until nearly absorbed, about 4 minutes. Repeat with 3 more cups stock, adding 1 cup at a time. Stir in kale and 1 cup stock; cook, stirring, until rice is tender and liquid is nearly absorbed, about 4 minutes. Thin with extra stock as needed. Off heat, stir in 3/4 cup cheese, the vinegar and the remaining butter; season with salt and pepper.
Meanwhile, in a skillet, heat 1 tbsp. EVOO over medium-high. Add half the mushrooms, season and cook until browned, 10 minutes; transfer to a plate. Repeat with remaining mushrooms. Divide risotto among plates; drizzle with EVOO. Top with mushrooms and remaining cheese.