To make the beer cheese sauce, heat a small saucepan over medium heat and add butter. Once sizzling, whisk in flour and stir for 1-2 minutes until golden and fragrant and bubbly. Add in beer and milk, whisking well to combine, then reduce heat to low and add cheese, stirring until totally melted and smooth. Keep over low heat and stir every few minutes while assembling sandwiches.
In a bowl, combine the chicken, wing sauce, ranch dressing, melted butter, green onions and gorgonzola, and stir to combine.
Heat a large skillet or griddle over medium heat. Spread the outsides of the bread with softened butter. To assemble the grilled cheese, place a slice of bread butter-side down in the skillet. Add on a few handfuls of fontina and cheddar cheese, then top it off with a quarter of the chicken mixture. Add a few spoonfuls of the melty beer cheese. Add more grated fontina and cheddar on top, followed by another slice of bread, butter-side up. Cook until the cheese is melty and each side is golden, about 4 to 5 minutes per side. Serve immediately.