Melt the butter in a fondue pot and add the shallots, garlic and thyme. Chop the morels, leaving the smaller ones whole, and add gently cook them. Remove the pot from the heat and let cool.
Add the fondue mix. Blend the cornstarch, wine and lemon juice and add it to the cheese. Stir constantly and bring to a boil. Lower the heat and simmer, stirring constantly until the cheese has melted. Add the cognac and seasonings.
To serve, but the burner on the table and let the fondue simmer over a low heat. Your guests can dunk the bread cubes into the sauce. To keep the cheese from sticking to the pot and turning bitter, stir the mixture with the bread cubes, taking care to scrape the bottom of the pot.