Heat 1 teaspoon oil in a skillet over medium-high heat. Season zucchini with salt and pepper and cook for 2 to 3 minutes per side until lightly browned and tender. Transfer to a plate and let steam escape for 2 minutes.
Spread pesto thinly over the roll cut faces. Divide half the Fontina over bottom halves, add folded turkey and zucchini in even layers and top with remaining cheese. Close the rolls.
Brush outer bread surfaces with remaining oil. Heat a large skillet over medium-low heat and place sandwiches in it with a clean heavy skillet set gently on top.
Cook for 4 to 5 minutes per side, removing the weight to turn, until bread is crisp, cheese melts and the filling reaches 165°F. Lower the heat if the crust browns before the center warms.
Rest for 1 minute and cut each panini across the middle before serving.