Make the turkey burgers: In a bowl, combine turkey, onion, chives, parsley, mustard, Worcestershire and poultry seasoning. Form into 4 thin patties. In a skillet, heat 1 tablespoon EVOO. Add the patties and cook, turning once, for 8-10 minutes.
Make the portobellos: In another skillet, heat the remaining 3 tablespoons EVOO over medium-high heat. Add the mushrooms and brown for 10 minutes. Add the shallot, thyme, salt and pepper; cook to soften, 5-10 minutes. Deglaze with sherry.
Wipe out the skillet and place over medium-high heat. Spread butter on 1 side of each bread slice. Layer each of 4 bread slices, buttered side down, with a cheese slice, a burger, some mushrooms, another cheese slice and another bread slice, buttered side up. Fry the melts in the skillet over medium heat until golden brown.