Combine dried mushrooms, garlic, thyme, and cream in a small saucepan. Bring to a boil; remove from heat, and let steep to allow flavors to meld for at least 30 minutes (or up to 2 hours). Pour through a sieve into a bowl; reserve mushrooms, and discard the garlic and thyme. Stir stock into cream mixture.
Preheat oven to 350 degrees. Pulse bread in a food processor until coarse crumbs form (you should have about 2 cups). Transfer to a bowl. Stir in 2 tablespoons oil, 1/2 teaspoon salt, and a pinch of pepper. Spread onto a rimmed baking sheet. Bake, stirring once or twice, until golden brown, about 25 minutes. Let cool. (Raise oven temperature to 375 degrees.)
Melt butter with remaining 2 tablespoons oil in a saute pan over medium heat. Add leeks, reduce heat to medium-low, and cook, stirring occasionally, until just tender, 8 to 10 minutes. Add fresh mushrooms, and raise heat to medium-high; season with 3/4 teaspoon salt. Cook, stirring often, until softened, about 7 minutes. Stir in reserved mushrooms. Remove from heat.
Bring a large pot of water to a boil; add 2 tablespoons salt. Working in batches, cook pasta squares, stirring occasionally to separate, until al dente, about 2 minutes. Using tongs or a slotted spoon, transfer to a colander to drain. Drizzle with oil, and toss gently to coat.
Spoon about 1 tablespoon cream mixture into each of 6 ovenproof dishes (about 6 inches in diameter, 1 1/2 inches deep). Place 1 pasta square in each dish, and sprinkle with about 2 teaspoons Parmigiano-Reggiano. Top each with about 1 1/2 tablespoons mushroom mixture, about 1 tablespoon cream mixture, and a pasta square. Repeat, layering two more times. Sprinkle top with about 1 tablespoon Parmigiano-Reggiano and about 1 tablespoon plus 1 teaspoon breadcrumbs, and drizzle with oil. Bake until cream mixture is bubbling and top is golden brown, 12 to 14 minutes.