Remove the bone and butterfly your turkey breast, skin side down (See notes below). Then cover with parchment or plastic wrap and pound it to a fairly uniform thickness.
Beat the eggs well in a small bowl with a dash of water and a bit of salt and pepper. Heat a small skillet with about a teaspoon of the olive oil and pour the eggs in the hot pan. Make sure egg covers the entire skillet bottom and let it cook for a few minutes, then flip it (you can use a plate to slide it on and turn it over into the skillet) to cook the other side briefly. Slide it back onto the plate and let it cool.
Season your pounded turkey breast with salt and pepper and begin laying the proscuitto and cheese on top, leaving about an inch all the way around without stuffing so rolling will be easier. Lay the torn herb leaves on top of that. Finally, add the omelet, cutting it, if needed, to fit it on top of the ham and cheese.
Begin rolling the turkey breast, starting on the short side, moving slowly and pressing well as you go, being careful to keep the filling inside.
Tie the turkey breast with kitchen twine on the short side about 1-2 inches apart. I also tied it once lengthwise, folding the meat and skin down and tucking under. Place a few sprigs of rosemary under the twine.
Pour the remaining olive oil into a large pan or dutch oven (one that can hold your turkey breast) and heat it over medium heat.
Once your oil is hot, place turkey roll in the pan, skin side down. Sear the meat over medium heat, without disturbing it, until it becomes golden brown. Turn it with tongs or two wooden spoons and brown on all four sides. This should take about 10 minutes.
Once it is brown on all four side, pour the 1 cup of white wine over it, letting it sizzle and evaporate for just a few seconds before you cover it with a tight lid. Turn the heat down to very low and cook for about 25-30 minutes, checking on it periodically to be sure it is not drying out. It should be fine if you have a tight lid and you keep it on very low heat.
Turn off the heat and just let the turkey rest in the pot, covered, for at least an hour. This is so the residual heat will continue cooking the turkey without drying it out. Don't skimp on this time or the turkey will not be cooked properly.
At this point you can remove the turkey breast and put it on a cutting board to slice. It is easier to slice the colder it is, so only cut what you plan to eat immediately and slice as you need it later or the next day (or two!)
To make the gravy, sprinkle the 2 tablespoons of flour over the pan drippings and whisk it in over medium heat. Once it is incorporated and bubbling, slowly add the broth, continuing to whisk as you go. Let this cook for a few minutes until it thickens.
To serve, warm the plates and pour a bit of gravy onto the plate, then place your sliced turkey and any sides on the plate, then pour more gravy over it.
I served this with simple mashed potatoes and a spring mix of shelled peas, snap peas, and spinach. In the fall you may consider mashed sweet potatoes or winter squash, and sauteed greens.