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Place a pot over medium heat and melt butter.
Once butter has melted, add flour and whisk until no lumps remain.
Whisk milk, mustard and salt into mixture and continue to whisk together until mixture is smooth and no lumps remain, and mixture has slightly thickened.
Remove mixture from heat and stir in grated cheese, small handfuls at a time, until all cheese has been added and cheese sauce is smooth.
Stack a third of the roast beef onto the bottom bun of each burger. Pour a generous amount of cheese sauce over each sandwich and gently press bun tops onto each sandwich. Serve immediately.