Warm the olive oil over medium heat in a 12-inch nonstick frying pan. Add the sliced onions along with a good pinch of salt. Cook, stirring often, until the onions are soft, lightly brown, and reduced to about 1/3 of their raw volume. You may need to reduce the heat gradually as the onions shrink. This process will take about 30 minutes.
Meanwhile, in a small bowl, toss the strawberry slices with 1 teaspoon of the balsamic vinegar. Wash the dandelion greens in several changes of water and dry thoroughly. Cut into bite-sized pieces if you like. (I like the shape of the leaves and will often just tear them in half to preserve it a bit.)
When the onions are nearly done, add the remaining 1 tablespoon + 2 teaspoons balsamic vinegar to the pan and continue cooking until it has thickened to coat the onions, a minute or two. Remove the onions from the heat and stir in the Dijon mustard.
In a salad bowl, combine the greens, onions and strawberries with all their juices, scraping down the onion pan with a rubber spatula to incorporate all the oil and vinegar into the salad. Sprinkle with salt and pepper and toss.