I recommend using a griddle to prepare the entirety of the burger. it just seems more succinct to me. You can cook the buns, burgers, and vegetables all in one place, and only dirty one appliance/pan. That said, feel free to turn on the grill and grill off the burgers. (I plan on writing a post dedicated to the advantages of griddles in certain situations, but for now just do what feel is right.) You can also sauté the vegetables with a pan in two batches as well.
The one thing you do need to do is use a mandoline on the squash and carrots. The carrots must be cut in a julienne because they are rich in pectin, and therefore require a generous distribution of heat to become somewhat soft. The situation is a little different for the squash since they contain a lot of water. I only mandoline the skin from the zucchini and summer squash because this is the only part of the squash that wont become mush once heated. Also, there will be a lot of leftover squash, so I recommend cubing, freezing, battering, and then frying the leftovers.
Dave's Summer Burger:
3 small summer squash
3 small zucchini
3 small carrots
1 tablespoon butter, melted
1 tablespoon olive oil
1 pound 80/20 ground beef
4 whole wheat buns
4 tablespoons butter, room temperature
1/2 pound fresh mozzarella cheese
Thinly sliced iceberg lettuce
8 slices of Spanish onions
1 thinly sliced tomato
Salt and pepper
1. Julienne the vegetables on a mandoline while you preheat a griddle or sauté pan. (400F for a griddle.)
2. Toss the vegetables in the oil and melted butter.
3. Cook the vegetables on the griddle until take on a medium-soft stiffness, about 5 to 7 minutes, moving the vegetables around as needed. Reserve while you cook the burgers.
4. While the vegetables cook make four evenly shaped patties and season them with salt and pepper, then butter the tops and bottoms of each bun.
5. Toast the buttered side of the buns and cook the burgers simultaneously.
6. Once the burgers are cooked and the buns are toasted, assemble the burgers with the rest of the toppings and let it rest for a minute or two so that the cheese loosens up.