Prep the vegetables. Toss the zucchini, squash, red pepper, and red onion with olive oil, salt, and pepper.
If you like, add a splash of balsamic to the peppers and onions for sweetness.
Grill or roast the veggies. Use a grill pan, outdoor grill, or a 425°F (220°C) oven. Cook until the vegetables are tender with light char, about 6–8 minutes on a grill or 12–15 minutes in the oven. Set aside.
Warm your cooking surface. Preheat a panini press, large skillet, or griddle over medium heat.
If you’re pressing in a skillet, have a second heavy pan ready to weigh down the sandwiches.
Build the panini. Spread pesto on the inside of each bread slice. Layer cheese on one side, then add the grilled veggies and a small handful of spinach or arugula. Top with the second slice, pesto side down.
Brush and press. Lightly brush the outside of each sandwich with olive oil or softened butter.
Place on the hot surface. If using a skillet, set the second pan on top to press.
Cook until crisp and melty. Grill 3–5 minutes per side, or 3–6 minutes total in a press, until the bread is deeply golden and the cheese has melted.
Rest and slice. Let the panini rest for 1 minute so the cheese settles, then slice in halves. Serve immediately.