The cheese that Panera uses for grilled cheese is American Cheese for making the classic Panera grilled cheese.
On the Panera Kickin Grilled Cheese and other varieties use a blend of American cheese and aged white cheddar cheeses or even a shredded mix of the 4 cheeses depending the menu item and promotions.
The best cheese to melt on toast is Cheddar cheese.
I prefer cheddar cheese when melting it on my toast as it has a good flavor and melts really good as well.
Other good cheeses to melt on your toast are Fontina cheese, Havarti cheese, Gruyere cheese and Monterey Jack cheese.
If you want a milder tasting cheese that also melts great you can use American cheese or even young Gouda cheese.
If you want a more bolder and more complex flavor of cheese on your toast along with a great melt, then using a combination of sharp cheddar cheese as well as a nutty and savory Alpine cheese like Comte cheese or Gruyere cheese are great choices of cheese to melt on toast.
Medium sharp cheddar cheese is also great and brings a richer flavor for melting on toast.
Young soft or semi soft cheeses like Swiss cheese, Havarti cheese or Gruyere cheese are great for melting on toast and have a high moisture content that means they are especially adept at turning molten.
Even Pepper Jack cheese is great for melting on toast and gives the toast a spicier twist.