PREHEAT OVEN TO 325° F
Coat an 11×7 baking dish with softened butter and set aside.
Place a nonstick frying pan over medium heat and add 1 T butter. When hot, add onions and saute until soft, about 4-5 minutes. Add garlic cloves and cook until fragrant, about 1 more minute.
Remove from heat and add 1 T fresh thyme. Set aside.
Whisk eggs, half and half, mustard, hot sauce and salt in large bowl until combined.
Arrange half of the Brazi Bites in the bottom of the greased dish.
Sprinkle with half of the onion mix, ham and Gruyere. Repeat layering with remaining Brazi Bites, onion mix, ham and Gruyere.
Pour egg mixture over the top and press down to immerse bread mixture in the egg mixture. Cover with plastic and place in refrigerator overnight.
Let bread pudding set at room temperature for 30 minutes while oven heats. Bake in middle rack for 20 minutes.
Sprinkle Parmesan over pudding and return to oven at 375 for another 15-20 minutes until puffed and pulling away from the sides.
Sprinkle remaining 1 T Thyme over the top. Cool for 5 minutes and serve.