Place the potatoes in a medium pot and cover with cold water. Add 1 teaspoon salt. Bring to a boil then reduce to a simmer and cook until the potatoes are fork tender but not falling apart, about 6-8 minutes.
Meanwhile, rinse the red onions with cold water and place them in a large bowl with the garlic. Add 1 tablespoon Sherry vinegar and a pinch of salt and pepper. Once the potatoes are cooked, drain them and immediately add the hot potatoes to the bowl with the onions; toss well. Stir in the dandelion greens and olives.
In a small bowl, whisk the remaining 1 tablespoon Sherry vinegar with the grainy mustard, honey, 1 teaspoon chopped tarragon and 3 tablespoons extra virgin olive oil. Pour the dressing over the fingerling potato and dandelion greens salad and toss to coat. Taste and season with salt and pepper as needed.
Season each chicken breast on both sides with salt and pepper. Sprinkle the remaining chopped tarragon evenly over each at pat to adhere (both sides).
Heat a large skillet over medium-high heat and add 1-2 tablespoons olive oil to coat. Working in batches, sauté each breast until golden brown on each side, 2-4 minutes per side. Add additional oil to the pan as needed, and cover the cooked chicken as it comes out to keep warm. (You can also grill the chicken—before grilling brush the chicken on both sides with oil.)
To serve, place a chicken breast on each plate. Mound the fingerling potato and dandelion greens salad over top. Crumble a nice amount of goat cheese over each, and serve.