In a large bowl, add the panko, milk, egg, sage, thyme and salt and pepper. Combine and let sit for a few minutes. Add in the turkey and combine well. Form into oblong patties (same size as the rye bread) Place on a parchment paper lined sheet pan and refrigerate for 20-30 minutes.
While burgers are resting, in a large saute pan, heat 1/4 cup of the butter and 1 Tablespoon canola oil. Add onions and saute 15 to 20 minutes or until nicely caramelized. Set aside, but keep warm.
In the same pan as the onions sauteed, add 3 tablespoons canola oil and the mushrooms and saute 6-8 minutes over medium high heat until softened. Set aside, but keep warm.
Heat a griddle or cast iron skillet to medium high heat, add 1 Tablespoon canola oil and spread evenly. Add in the turkey burgers and cook for 4 to 5 minutes per side. Remove from heat, cover and keep warm.
Butter the rye bread and add to the pan, place the gruyere on bread, then place the turkey burgers on the cheese, pile with onions and the other piece of rye, repeat for all four burgers. After 2 to 3 minutes, turn sandwich over carefully and brown on other side. Remove to cutting board and cut on the diagonal. Serve with mustard and hot cherry peppers.