Savory French Toast

INGREDIENTS

• 5 ounces (about 6 slices) bacon
• 2 onions, cut into 1/4-inch dice
• 12 ounces button mushrooms, stemmed and quartered
• 6 large eggs
• 1 1/2 cups heavy cream, half-and-half, or milk
• 1/2 cup grated Parmesan cheese (about 1 1/2 ounces), plus more for serving
• 1 teaspoon coarse salt
• 6 slices 1-inch-thick bread, such as sourdough, each cut into 3 long strips
• 2 tablespoons unsalted butter
• 2 tablespoons vegetable oil