Separate the sliced shallots into individual rings and set aside. In a small saucepan, combine 1 teaspoon of the olive oil with the chopped shallot and cook over medium heat, stirring occasionally, for about 3 minutes, until limp. Add the juice, increase the heat to medium-high and boil until reduced to 1/4 cup, 10 to 12 minutes. Stir as the liquid becomes syrupy to prevent burning. Remove from the heat and set aside to cool.
Add about 1 inch of vegetable oil to a medium saucepan and place over medium heat. In a shallow bowl, toss the shallot rings with the flour, then shake in a mesh strainer to get rid of any excess flour. When the oil is hot (it will ripple), add the shallot rings, stirring so they don't stick together. Fry until crisp and brown, 2 to 4 minutes. Remove with a slotted spoon, drain on paper towels and season with salt.
When the pomegranate sauce is cool, pour it into a medium bowl. Whisk in the vinegar and a few dashes of salt and pepper, then slowly whisk in the remaining olive oil. Snag a salad leaf and dip it into the dressing to taste. Adjust the salt and pepper.
Place the greens in a large bowl and dress lightly (you might not need all of the dressing). Divide the greens among 6 plates and top with the fried shallots and pomegranate seeds.