Poach the eggs: Add vinegar to a large saucepan of simmering water. Crack eggs into water, 1 at a time. Poach eggs until whites are set but yolks are still runny, about 2 minutes. Transfer to a paper-towel-lined plate using a slotted spoon.
Make the salad: Heat 2 tablespoons oil in a high-sided skillet over medium heat. Cook potatoes until golden, about 6 minutes per side. Transfer to a large bowl using a slotted spoon, and add spinach. Add remaining tablespoon oil to skillet. Cook chorizo until golden brown, about 3 minutes per side. Add to bowl using slotted spoon.
Make the dressing: Remove skillet from heat. Add shallot, and let cook for 30 seconds. Swirl in vinegar.
Pour dressing over salad, season with salt and pepper, and toss to coat. Divide salad among 4 plates. Top each with an egg.