In a small bowl, cover the sun-dried tomatoes with very hot tap water. Let stand while you cook the squash.

• Trim the ends from the summer squash; cut the squash on the diagonal into ¼-inch-thick slices.

In a large frying pan over medium-high heat, warm 2 tablespoons oil until hot but not smoking. Working in batches if needed, arrange the squash in a single layer, season with salt and pepper, and cook, turning once, until lightly browned and just starting to soften, 2 to 3 minutes per side. Transfer to a medium bowl. Wipe out the pan.
While the squash cooks, prepare the remaining panini ingredients.

• Cut the mozzarella into ¼-inch-thick slices.
• Strip the basil leaves from the stems.
• Drain the sun-dried tomatoes and pat dry; thinly slice the tomatoes.
• Thinly slice the mushrooms.

Place the flatbreads side by side. Top half of each flatbread with mozzarella, sun-dried tomatoes, basil leaves, mushrooms, and arugula, dividing evenly, then fold the flatbread in half.
Warm the pan used for the squash over medium heat. Brush the outside of the panini lightly with oil, add to the hot pan, and cook over medium heat until the flatbread is toasted and the mozzarella has melted, 2 to 3 minutes per side.
While the panini cook, finish the squash.

To the bowl with the squash, add the pumpkin seeds and arugula pesto and toss to coat; season to taste with salt and pepper.

Transfer the panini to individual plates and serve the squash alongside.