In a small bowl, combine the shallots, salt, and vinegar. Set aside while you peel the oranges.
Slice off the end of each orange. Squeeze the juice from each end into the bowl of shallots and vinegar. Working with one orange at a time, stand it on its flat end. Use your knife to slice down around the orange to remove the skin. Hug the flesh as you slice—it’s better to lose some flesh in the rind than to have pith clinging to the flesh.
Squeeze the rinds into the bowl of shallots and vinegar to extract any juice. Repeat until you’ve peeled all of the oranges.
Working with one orange at a time, turn on its side and slice down crosswise to create 1/2-inch thick slices. Arrange the oranges on a platter—season with flaky sea salt.
Using a fork, whisk the olive oil into the dressing. Taste. Add salt if necessary or more oil for a little less bite. Drizzle a few spoonfuls of dressing over the oranges.
Place the greens and thinly sliced fennel into a large bowl. Drizzle with a few spoonfuls of the dressing. Toss lightly to coat. Taste. Adjust with more dressing or salt to taste.
Pile the greens on top of the oranges.
Use a peeler to shave Parmigiano Reggiano over top.
Crack pepper to taste over the top. Serve with knife and fork. Store extra dressing in a jar in the fridge or at room temperature if your kitchen is cool.