Slice potatoes on mandolin to 1/4-inch-thick disks. Select 60 disks of at least 1 1/2 inches in diameter; reserve remaining pieces for other use. Par-cook disks for 2 minutes in boiling water; drain well. Arrange in a single layer on baking sheet to air-dry and cool, about 5 minutes. (Do not shock in cold water.)
Prepare Herbed Goat Cheese:
Mix all ingredients together in bowl with wooden spoon; reserve.
To assemble each of 12 Napoleons, place a potato slice on a work surface; top with a teaspoon of Herbed Goat Cheese. Continue layering: a potato slice, a teaspoon of cheese, a potato slice, a teaspoon of cheese, a potato slice, a teaspoon of cheese and finish with a potato slice. Gently press and compact so Napoleon measures about 2 1/2 inches in height and cheese is neatly contained. Repeat to make a total of 12 Napoleons. Transfer to clean sheet pan; refrigerate 20 to 30 minutes, or up to 1 day.
Prepare Heirloom Tomato Confit: Place tomatoes, olive oil, garlic, thyme sprig, 1/4 teaspoon salt and sugar in shallow baking dish. Bake 1 hour in preheated 250°F oven. Remove tomatoes from oil, reserving oil. Cool tomatoes; discard garlic and thyme.
In a stainless mixing bowl, whisk together 1/4 cup of rice vinegar and oil reserved from tomatoes. Add cooled tomatoes, basil, parsley, shallots, pepper, remaining 1/4 teaspoon salt and chopped thyme; mix well. Reserve at room temperature, covered, up to 1 hour.
Remove Napoleons from refrigerator. Carefully dip each in flour, shaking off excess. Dip in egg wash then gently roll in panko crumbs, pressing crumbs between the layers.
In deep, medium-gauge saucepan, heat oil to 300ºF. Working in batches of four, carefully place Napoleons in oil; cook until golden, about 5 minutes. Transfer with slotted spoon to paper towel; reserve warm.
Divide Heirloom Tomato Confit among four plates. Carefully cut each Napoleon in half; place 6 pieces per plate on top of confit, arranging cut-side-up in desired plate presentation. Sprinkle each piece with salt, pepper and 1/2 teaspoon Parmesan. Scatter 1 teaspoon basil over tomatoes on each plate.