How to Make It
Break matzos into pieces and place in a colander. Moisten under cold running water for 15 to 30 seconds. Transfer matzos to a large mixing bowl.
In a large nonstick skillet, melt 1/2 Tbsp. butter with 1/2 Tbsp. oil over medium-high heat. Add onion; season with salt. Cook, stirring often, until softened, about 5 minutes. Add mushrooms. Season with salt, then cook, stirring, until mushrooms soften, 3 to 4 minutes longer. Stir in thyme.
Scrape mushroom mixture into bowl with matzo. Stir in eggs and cheese, if desired. Season with salt and pepper.
Wipe skillet clean. Melt remaining 1/2 Tbsp. butter with 1/2 Tbsp. oil over medium heat. Pour matzo mixture into skillet and spread into an even layer. Cook until golden brown on the underside, 10 to 12 minutes, occasionally loosening it from bottom of pan with a thin spatula.
Place an oven rack 6 inches from broiler. Preheat to high. Slide skillet under broiler; cook until top of matzo brei is golden, 1 to 2 minutes, watching carefully to prevent burning. Slice into wedges and serve immediately.