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