Kosher salt
8 ounces green beans, cut into 1-inch lengths
Freshly ground black pepper
2 tablespoons olive oil
1 tablespoon unsalted butter
1 shallot, thinly sliced
3 large Portobello mushrooms, stems and gills removed
8 ounces shiitake mushrooms, stems removed and caps thinly sliced
2 teaspoons fresh thyme leaves
1/2 cup dry white wine
2 tablespoons all-purpose flour, plus more for dusting
3/4 cup half-and-half
2 tablespoons plain breadcrumbs
1 sheet frozen puff pastry (half a 17.3-ounce package), thawed
2 teaspoons Dijon mustard
1 large egg, beaten
Large-flake sea salt, such as Maldon