Peel potatoes and zucchini and grate onto a large kitchen towel using a box grater.
Wrap grated zucchini and potato in your kitchen towel and squeeze as much liquid out as you can, into your sink.
Remove dried zucchini and potato and place into a large bowl.
Add the egg, lemon juice and zest, tarragon, salt, parmesan cheese, cheddar cheese, garlic, and Italian and panko breadcrumbs and mix until combined.
Heat your cooking oil in a large sauté pan over medium-high heat. Measure your latkes in 8 pucks, placing a heaping spoonful or two of mixture into a ring mold.
Place your latkes in the hot oil using a spatula larger than the ring mold, bracing the latke from underneath. Carefully slide each latke into the hot oil and remove ring mold with tongs to keep from burning your hands.
Cook each latke for 3-5 minutes on a side until deeply golden brown and crispy. Flip carefully and cook 3-4 minutes more on the second side.
Transfer latkes to a paper towel-lined baking sheet and season with a little salt to taste. Serve with a dollop of sour cream and basil oil (1 cup of fresh basil, cooked and dried, blended with 1 cup olive oil).
If making your own basil oil, strain through a cheesecloth after blending to remove cloudiness and basil sediment. Store in an air-tight squeeze bottle.