Melt 3/4 tablespoon butter with 3/4 tablespoon olive oil in a skillet over medium heat. Cook bread cubes in the butter mixture, tossing gently to coat, until fragrant and toasted, 3 to 4 minutes. Add garlic to the bread; cook and stir together for 1 minute. Remove to a plate and set aside to cool.
Slice zucchini lengthwise with stem and flower ends intact. Use a small spoon to hollow out squash, leaving a 1/4-inch shell. Place squash boats in a large baking dish. Coarsely chop zucchini innards.
Melt 3/4 tablespoon butter with 3/4 tablespoon olive oil in a skillet over medium-high heat. Cook and stir zucchini innards, tomatoes, rosemary, salt, and pepper in the butter mixture until the vegetables soften, 3 to 4 minutes; transfer to a large bowl and mix with the bread cubes, Cheddar cheese, and beaten egg. Fill the hollowed zucchini halves with the vegetable mixture. Arrange stuffed zucchini boats into a baking dish.
Bake in preheated oven for 25 minutes. Remove and switch oven to broil. Cook under the broiler until golden brown and bubbling on top, 2 to 3 minutes.