Preheat oven to 400 F.
In a medium-sized bowl, whisk the milk, broth, eggs, salt, pepper, tarragon, and rosemary until combined.
In a large saute pan on medium heat, heat the olive oil.
Cook the onions and garlic for two minutes until softened.
Add the zucchini and squash and let sit to brown. Then, turn occasionally, cooking for 5-6 minutes until soft.
Set the stale bread into the milky mixture and fold to coat.
Mix in the zucchini/squash mixture and stir to coat.
Spray a 9"x13" baking dish with nonstick cooking spray (I use canola oil spray) or grease with butter.
Pour bread pudding into baking dish.
Sprinkle cheese evenly over the top of the dish.
Bake for about 40 minutes until browned and liquid is absorbed.
Let sit for 10 minutes to solidify.
Serve directly from the baking dish or transfer to shot glasses.