Fill a large, deep skillet with about 3 inches of water and stir in the white vinegar. Warm over medium-high heat until the water is just below a boil and small bubbles are rising and popping across the surface.

Crack the first egg into a fine-mesh sieve set over a bowl so the thin, watery part of the white drains away, then gently slide the egg into a small bowl. Repeat with the remaining eggs.

Carefully tip each egg into the hot water one at a time, leaving space between them so the whites do not touch. For best results, poach only 4 eggs at a time.

Cover the skillet and set a timer for 3 minutes and 30 seconds for large eggs.

Using a slotted spoon, lift the eggs from the water in the same order they were added. Gently press the white to check for doneness; if you prefer firmer eggs, return them to the water for 15 to 30 seconds. Transfer to a paper-towel-lined plate and keep the poaching water on low heat for rewarming before serving.

Pour about 1/4 inch of vegetable oil into a 12-inch cast iron skillet or another large heavy skillet. Heat over medium heat to 350°F, or until a wooden skewer inserted into the oil starts to bubble steadily. If it bubbles furiously, reduce the heat and let the oil cool for a few minutes.

Carefully place 4 frozen hash brown patties in a single layer in the skillet. Fry for 8 to 10 minutes, flipping halfway through, until golden brown and crispy. Use a slotted fish spatula to transfer them to a paper-towel-lined plate and season with salt. Repeat with the remaining 2 patties, adding more oil if needed.

Rewarm the poached eggs by dipping each one into the reserved hot water for 30 to 60 seconds, then place them back on the paper-towel-lined plate and season lightly with salt and pepper.

Arrange 2 hash brown patties on each plate. Cut the avocados in half, remove the pits, and scoop the flesh into a bowl. Mash with a fork along with the lime juice, salt, and pepper until creamy but still a little chunky.

Spoon a generous scoop of the mashed avocado onto each hash brown patty. Place a poached egg on top, finish with salt, pepper, and chopped chives, and serve right away. Pass the optional toppings at the table so everyone can customize their own toast.