Soften the morels in water, rinse well and drain. Finely chop the shallots and thyme, press the garlic. Heat the butter in a fondue pot, sauté the shallots, garlic and thyme. Roughly chop the large morels and add the smaller ones whole, cook briefly, remove the fondue pot from the heat, leave to cool.
Add the fondue mixture, stir in the cornflour. Pour in the champagne and lemon juice, bring to the boil while stirring, reduce the heat, continue to simmer and stir until the cheese has melted. Add the cognac, season.
Keep the fondue simmering on a low heat on a burner at the table.