In a small bowl, whisk together 6 tablespoons peanut oil and vinegar; season with salt and pepper. Set aside.
Heat 1 cup peanut oil in a deep skillet until it registers 350 degrees on a deep-fry thermometer. Soak onion rings in milk for 5 minutes; drain, and spread onions on a baking sheet. Whisk together flour, paprika, and a pinch of salt in a small bowl. Sprinkle onions lightly with seasoned flour. Carefully add onions to skillet, and fry until just golden and crisp. Transfer to a paper-towel-lined baking sheet to drain.
In a small heavy saucepan, add enough oil to fill it 2/3 full. Heat oil over medium heat until it reads 350 degrees on a deep-fry thermometer. Working with one egg at a time, break egg into a small bowl, and carefully slide it into the hot oil. Cook, using two wooden spoons or spatulas to shape the egg and maintain a rounded shape, for 1 minute; carefully turn over egg. Cook until egg is crunchy on the outside and yolk is still runny, 1 1/2 to 2 minutes. Using a slotted spoon, lift egg out of oil, and transfer to paper-towel-lined baking sheet. Repeat with remaining eggs.
In a medium bowl, toss together dandelion leaves, reserved dressing, and shallots. Divide between 4 serving plates, and top each portion of salad with a fried egg. Garnish with fried onion rings, and serve immediately.