Note: You'll need to make the pesto (mentioned under "Other Toppings") before grilling the chicken and flatbreads. My homemade pesto only takes a few minutes to make. You can also use store-bought pesto.
Prepare and Grill the Chicken and Flatbreads:
For the chicken, rinse and pat the chicken breast(s) dry. In a shallow and wide container, or a Ziploc, whisk (or shake) the marinade ingredients together: olive oil, lemon juice, red wine vinegar, dried oregano, minced garlic, salt and pepper.Place the chicken in the marinade, and toss the chicken around so it's covered. Marinate the chicken in the fridge for at least 30 minutes, up to 12 hours. The chicken will appear to be "broken down" if you choose to marinate for 12 hours due to the acid, but the taste and texture don't reflect this once cooked.
Preheat an outdoor grill to roughly 450F (preheated for about 10-15 minutes on a high flame).
Remove the chicken from the marinade (toss the extra marinade in the trash), and cook the chicken directly over the heat (on the grill) until it's brown on the side that's facing the heat. Once brown, flip the chicken and continue to cook on the other side until brown. The time it takes for your breast(s) to cook will vary depending on the size of the breast(s). I usually cook the chicken for about 10 minutes on each side. The chicken is done when a meat thermometer registers at 160F.
While the chicken cooks, prepare and grill the flatbreads. Dust a large sheet of parchment paper with einkorn flour. Divide the dough into 4-6 small balls, and then roll each dough ball out to create a small flatbread. Brush a small amount of olive oil over the top of each flatbread.
Work quickly to place the dough on the grill. Cook the flatbreads for about 3 minutes, with the grill covered, until the bottom of the dough easily lifts away from the grill and grill marks are apparent. Use a heavy spatula to get under the flatbreads and flip them. Cook the flatbreads, with the grill covered, for another 2-3 minutes. Each grill differs slightly, just like ovens, so keep a close eye on the flatbreads so they don't burn.
When the chicken is done cooking, let it rest for 5-10 minutes before slicing.
Make the Arugula Salad:
In a medium-size bowl, toss together the arugula, olive oil, lemon juice, and red wine vinegar.
Assemble the Flatbreads:
Spoon 1-2 tablespoons of pesto on each flatbread, followed by a small handful of the arugula salad, chicken, and toppings (cherry tomatoes, radishes, feta).