Brush a 9 x 11 in. (23 x 28 cm) baking dish with 1 tbsp (15 mL) oil and set aside. Heat remaining oil in a skillet over medium-high heat and sauté mushrooms and diced shallots until tender. Stir in fresh thyme, season with salt and pepper and cool slightly.
In a mixing bowl, toss sautéed mushrooms, cubed bread and 1/2 cup (125 mL) cheese; spread mixture evenly in dish.
Whisk together eggs, milk, paprika, cayenne, salt and pepper. Pour egg mixture over the bread mixture, wrap with foil and refrigerate for 3 to 4 hours or overnight.
Bake in an oven preheated to 350°F (177°C). After 20 min., remove foil and sprinkle strata with remaining shredded cheese; bake for an additional 20 to 25 min., until the top is golden and bubbly. Cool slightly; cut into squares and serve.