Bring stock to a gentle simmer in a saucepan over medium-low heat. Then turn off the heat, leaving the saucepan on the stove to keep it warm. Adding cold stock to the rice stops the cooking process and slows down starch release.
Heat 1 Tbsp olive oil and 1Tbsp butter in a wide skillet over medium-high heat. Add the garlic and thyme sprigs and sauté for 1 minute. Add the mushrooms and cook for 4 to 5 minutes until deep golden brown. Add the salt and pepper, and sauté for another few minutes until crispy and caramelized. Set roughly ¼ aside for garnish and leave the rest to fold into the rice later.
In a heavy-bottomed Dutch oven or deep skillet, melt 1 Tbsp butter over medium heat. Add the diced shallot and garlic and cook for 2 to 3 minutes until soft and translucent. Add the rice and stir continuously for 2 minutes until the edges of the grains become translucent and the centers look chalky white.
Pour in the white wine and stir constantly until fully absorbed by the rice, about 1 to 2 minutes.
Add the warm stock, 1 ladle (about ½ cup) at a time, stirring frequently. Allow each ladle to be almost entirely absorbed before adding the next. Continue this process for 18 to 25 minutes until the rice is tender with a slight bite (al dente). Stir in the caramelized mushrooms during the final 5 minutes, remembering to leave a few for garnish.
Remove from heat. Stir in the remaining 1 Tbsp butter and Parmigiano-Reggiano. Cover and let sit for 2 minutes so the emulsion settles into a glossy sauce. Scoop into serving dishes—the texture should flow slowly like lava rather than sit in a rigid mound on the serving dish. Top generously with the reserved caramelized mushrooms, extra grated Parmesan, and a fresh thyme sprig.