Bring a pot of salted water to a boil. Add the pasta and cook as the label directs, then drain.
Meanwhile, melt the butter in a large saucepan over medium heat. Add the shallot and thyme and cook until softened. Whisk in the flour and cook, whisking, for 2 minutes, then whisk in the broth and milk. Season with 1/2 teaspoon salt and a few grinds of pepper. Cook, whisking, until thickened, about 5 minutes.
Stir in the chicken, spinach and pasta and cook to heat through. Stir in the gruyère and season with salt and pepper. Divide among 4 bowls and top with toasted breadcrumbs.