Note on cooking the Beef Patties - My stove top is ridiculously hot. Every time I make burger patties I make them thin so they cook faster without burning. I used medium-high heat, because if I use high heat it would smoke up my pan. Try cooking the patty on medium high heat, and leave it a little longer if it's not cooked through - we're looking for a 160°F internal temperature for a well done, but moist burger patty here.
Note on Mayonnaise - If your food processor is too big to make 1 yolk mayonnaise, you can use a hand held beater with a whisk attachment. Make sure you whisk the egg and oil together in a deep and narrow "bowl" (I use a wide glass or jug), to make the mixture swirl naturally with the whisk making the whisking and mixing easier.