Start by preheating your oven to 375°F. Clean the mushrooms by gently wiping them with a damp paper towel. Avoid rinsing under water, as mushrooms soak up moisture and can get soggy. Using a paring knife, trace around the stems and pull them out. You can save these stems for homemade stocks or soups later.
Take each mushroom cap and stuff it almost to the top with your herb pub cheese. Make sure the filling is level and doesn’t overflow. A small spoon or piping bag makes this step neat and easy. Place the stuffed mushrooms on a parchment-lined baking sheet.
In a small bowl, combine the grated Parmesan cheese, chopped provolone, and half and half. Microwave this mixture for about 25 seconds until slightly melted. Stir in the panko breadcrumbs and let it sit for a few minutes. As the mixture cools, it will firm up, making it easier to handle.
Using your fingers, top each mushroom with a small portion of the cheese mixture. Gently press it down so it sticks, but be careful not to over-stuff. Too much filling may bubble over during baking and create a mess on the tray.
Transfer the tray of mushrooms into the oven and bake for about 15 minutes. Keep an eye on them—the tops should turn golden brown, while the mushrooms remain firm enough to pick up. If needed, rotate the middle mushrooms with outer ones halfway through baking to ensure even browning. Let them rest for 5 minutes before serving so they can firm up and cool slightly.
While the mushrooms are baking, start on the sauce. Grate the white cheddar and let it come to room temperature. Toss it with a tablespoon of flour to avoid clumping. In a heatproof bowl over a double boiler, combine half and half, wine, Worcestershire sauce, dry mustard, pepper, and mayonnaise (or sour cream). Heat gently for about 15 minutes, stirring occasionally.
Slowly add in the cheddar cheese and stir until smooth and creamy. Remove from heat once fully combined. Serve the sauce by drizzling it over the mushrooms, spreading it onto a plate Longhorn-style, or offering it in ramekins as a dip.