Cut an X in bottom of each tomato with a sharp paring knife and blanch tomatoes together in a 6- to 8-quart pot of boiling water 10 seconds.
Transfer tomatoes with a slotted spoon to a bowl of ice and cold water to cool. Peel off skin with paring knife, beginning from scored end, and discard. Finely chop tomatoes.
Peel potatoes and cut into slices slightly less than 1/4 inch thick.
Heat oil in a 4- to 5-quart pot over moderately high heat until it registers 370°F on thermometer. Fry potatoes in 6 batches, turning over occasionally, until golden, about 6 minutes per batch, transferring with slotted spoon to paper towels to drain. (Return oil to 370°F between batches.)
Heat butter in a wide 6- to 8-quart heavy pot over moderately high heat until foam subsides, then add onion and garlic and sauté, stirring occasionally, until golden, about 8 minutes.
Add beef and sauté, stirring occasionally and breaking up large lumps with a wooden spoon, until meat is no longer pink, about 6 minutes.
Add tomatoes, salt, and pepper and cook, stirring occasionally, until almost all of liquid is evaporated, 15 to 20 minutes.
Remove from heat and stir in cheese and bread crumbs until just combined.
Arrange one third of potatoes, overlapping slightly, in a buttered 13- by 9-inch glass baking dish, then spread with half of beef mixture. Repeat layering with potatoes and beef, ending with a third layer of potatoes.
Put oven rack in middle position and preheat oven to 375°F.
Whisk together butter, eggs, milk, cheese, and bread crumbs until just combined, then pour evenly over potatoes.
Bake until golden brown and bubbling around edges, 35 to 40 minutes. Cool slightly before serving.
