Pour 1/4 cup oil into a 9-by-13-inch glass baking dish. In a large bowl, whisk both flours with 2 tsp. salt. Put 1 1/2 cups warm water (110° to 115°) in a medium bowl. Add 3 tbsp. oil and the yeast to the water; whisk to combine.
Make a well in the flour mixture and pour in the yeast mixture. Using your hands, mix the ingredients until the dough comes together to form a slightly sticky ball. Place the dough in the oiled baking dish and cover with a damp towel. Let the dough rise at room temperature in a warm, draft-free area until doubled in size, about 2 hours.
Preheat the oven to 375°. Gently stretch the dough to fit the dish. Using wet fingertips, create divots in the surface of the dough. Drizzle the dough with the remaining 3 tbsp. oil and top with the cheese, shallots and thyme. Bake until the bread is golden brown, about 25 minutes. Let cool at least 10 minutes. Serve warm or at room temperature.