Preheat the oven to 400°F (200°C or 190°C for fan ovens).
First, make the crust. Melt the butter over medium heat.
Sauté the onion for two or three minutes until it begins to become translucent.
Grate in the garlic, and add salt and pepper to taste.
Sauté everything for another minute or two.
Remove from the heat and set aside to cool a bit.
Crack one of the eggs into a small bowl or ramekin and beat it with a fork until light and fluffy. Set aside.
Put the French bread cubes in a large bowl.
Add the onion mixture, stock and the lightly beaten egg to the bread cubes.
Stir to coat thoroughly with a wooden spoon and then continue to work the mixture together with clean hands so that all the bread cubes are moistened.
Press the bread mixture into the pie dish, working it up the sides.
Bake the crust for 10 to 15 minutes until the crust just begins to turn golden brown.
Remove the crust from the oven and set aside on a heat proof surface.
Lower the oven temperature to 375°F (190°C or 180°C for fan ovens).
Meanwhile, steam the broccoli, red pepper and carrot (or whatever fresh vegetables you wish to use) over lightly boiling water for 2 to 3 minutes. (If you are using leftover vegetables, you can skip this step.)
Drain the vegetables in a sieve and run some cool water over them to refresh them a bit. Drain again and set aside.
Beat the remaining four eggs with the milk in a medium mixing bowl.
Stir in the basil and three-quarters of the grated cheese.
Add the salt and pepper.
Arrange the vegetables evenly over the crust.
Carefully pour the filling over top of the vegetables, taking care not to over-fill the dish. (Depending on the size and depth of your pie dish, you may end up with a bit of extra filling.)
Sprinkle the top of the Crunchy French Bread Egg Pie with the remaining cheese.
Bake for 35 to 40 minutes, or until the pie is set and the top is beginning to turn golden brown.
Remove from the oven and allow to rest for about 10 minutes before slicing.