Heat 1 Tbsp. oil in a large skillet over medium heat. Add mushrooms and 1/4 tsp. salt; increase heat to medium-high and cook, stirring occasionally, until the mushrooms have released their liquid and turned golden, 8 to 10 minutes. Transfer to a bowl; cover and set aside.Advertisement
Meanwhile, working in 2 batches, pulse cauliflower in a food processor until it breaks down into rice-size pieces. You should have about 5 1/2 cups.
Wipe out the pan. Add the remaining 1 Tbsp. oil and reduce heat to medium. Add shallot and cook, stirring often, until tender, about 2 minutes. Add garlic; cook, stirring often, until fragrant, about 1 minute. Add the cauliflower "rice," thyme, 1/4 tsp. pepper and the remaining 1/4 tsp. salt; cook, stirring, for 2 minutes. Add wine; cook, stirring, until the wine evaporates, 1 to 2 minutes. Add broth; cook, stirring, for 1 minute. Cover, reduce heat to medium-low, and cook, stirring once or twice, until the "rice" is very tender, 4 to 5 minutes. Uncover, increase heat to medium, and add butter. Cook, stirring, until the butter has melted, about 1 minute longer. Stir in Parmesan, the reserved mushrooms, and the remaining 1/4 tsp. pepper.
Divide the mixture among 4 bowls. Sprinkle with parsley and serve with lemon wedges, if desired.