Pat the chicken dry with paper towels. Season with salt and pepper. In a large pan (nonstick, if you have one), heat the ghee on medium-high until melted. Add the seasoned chicken in an even layer. Cook, without stirring, 3 to 4 minutes, or until lightly browned. Continue to cook, stirring occasionally, 3 to 4 minutes, or until browned and cooked through. Turn off the heat.
While the chicken cooks, in a small pot, heat 2 teaspoons of olive oil on medium-high until hot. Add the tomato sauce (carefully, as the liquid may splatter) and Italian seasoning. Season with salt and pepper. Cook, stirring occasionally, 3 to 4 minutes, or until slightly thickened. Turn off the heat. Taste, then season with salt and pepper if desired.
Assemble the paninis using the halved rolls, pesto, cooked chicken, mozzarella (tearing into small pieces before adding), and the pecorino; season with salt and pepper. Wipe out the pan used to cook the chicken. In the same pan, heat a drizzle of olive oil on medium until hot. Add the paninis. Place a heavy-bottomed pot (or pan) on top and press down. Cook, occasionally pressing down on the pot, 4 to 5 minutes per side, or until the cheese is melted and the bread is lightly browned (if the pan seems dry, add a drizzle of olive oil before flipping). Carefully transfer to a cutting board and halve on an angle.