Select half of the carrots—choose the least attractive ones—and slice them into thick coins. Reserve the remaining carrots for the spiral surface of the tart. Place the cut carrots in a large saucepan with the onion and just enough broth to cover (you should have some broth left). Bring the broth to a simmer over medium-high heat, reduce the heat to medium-low, cover the pan, and gently cook the carrots for 30 minutes. Uncover and cook for an additional 15 to 20 minutes, or until the carrots are very tender and almost all of the liquid has evaporated. Let cool for 10 minutes.
Dump the cooked carrots and any liquid remaining in the pan into a blender and purée until smooth, adding an extra splash of broth if necessary to loosen the mixture just enough to puree it. Pour the purée into a skillet or sauté pan placed over medium-low heat. Cook briefly, stirring frequently with a wooden spoon or spatula, until the purée is thick and any liquid has evaporated. Remove from the heat and stir in 2 tablespoons of butter. Whisk in the egg yolks and Parmigiano cheese and season to taste with salt and pepper.
Cut the reserved carrots into thin, uniform coins. Melt 2 tablespoons of butter in a large skillet or wide sauté pan over medium-low heat. When the butter has melted, add the carrots and sprinkle with a little salt and pepper. Cook the carrots over medium-low to low heat for about 30 minutes, or until they're just tender, moistening them from time to time with a splash of broth to prevent them from sticking to the bottom of the pan. Remove the pan from the heat and let the carrots sit until cool enough to touch.