Savory Gruyère Stuffed French Toast

INGREDIENTS

• 2 tablespoons port or sherry
• a few grinds of fresh pepper
• 3 garlic cloves, smashed and peeled
• 1 teaspoon fresh thyme, minced
• 1 tablespoon butter, plus more for frying
• 3/4 pounds Gruyère, sliced
• 6 eggs
• 1 pound crimini mushrooms, sliced
• 1 loaf unliced challah bread
• 1 teaspoon minced sage
• 1/2 teaspoon salt, divided
• 1 1/2 cup whole milk
• 1 tablespoon canola oil