For the tomatoes

1. Prep tomatoes. Trim 1/2-inch from the stem and blossom ends of the tomato and discard. Slice tomato into 1/4-inch thick slices. Season tomato slices on both sides with salt and pepper. Set aside.
2. Prep coating. In a shallow dish or pie pan, add the flour. In a second shallow dish, whisk together the egg and buttermilk until blended. Spread the panko breadcrumbs in a third shallow dish.
3. Coat tomatoes. Working with one tomato slice at a time, dip the tomato slices into the flour to coat evenly, shaking off the excess. Then, dip into the egg mixture, letting the excess drip back onto the dish. Then coat the tomato slices in the panko breadcrumbs, patting firmly to help the breadcrumbs adhere.
4. Cook tomatoes. Heat 1/2 cup oil in a 10-inch non-stick or well-seasoned cast iron skillet over medium-high heat until it shimmers. Working in batches if necessary, carefully place tomatoes into the heated oil in a single layer. Cook until the undersides are golden brown, about 2 minutes. Turn the slices and fry until the other sides are browned, about 2 minutes more, adding more oil to the pan as necessary. Drain the fried tomatoes on paper towels.

For serving

1. Prep avocado. Cut the avocado in half lengthwise around the pit. Twist the avocado halves in opposite directions and pull the halves apart. Remove the pit from the avocado and discard. Scoop out the avocado pulp into a medium bowl and mash it with a fork and some fresh lemon juice. Season with salt and pepper to taste.
2. Plate tomatoes. Place tomato slices on plates and top with even amounts of avocado mixture. Set aside.
3. Cook eggs. For the eggs, crack the eggs into 4 small ramekins or sauce dishes. In a 12-inch straight-sided skillet, bring 3 inches of water to a simmer. Turn down the heat so that no more bubbles form on the surface. Carefully slide the eggs into the water, evenly spreading the eggs around the pan so they don’t touch each other. Cook eggs, undisturbed, until white is just set and yolk is still runny, 3 to 4 minutes. Use a rubber spatula to gently release eggs from the bottom of pan, if necessary. Using a slotted spoon, remove eggs from water. Blot the bottom of the spoon on a paper towel to remove excess water. Gently place an egg on each tomato slice. Serve immediately.