Bring a medium size skillet up to heat over medium high. Add in the butter and EVOO. Once melted add in the onions and reduce heat to medium low. Season with salt and pepper. Cook the onions until they are golden brown and caramelized. Remove from heat.
In a bowl whisk together the eggs, heavy cream, rosemary and a pinch of salt and pepper.
Slice the baguettes into 1/2 inch rounds. If they still feel a little soft you can quickly toast them in the oven to dry them out. Shingle them in an even layer in a heavy bottomed round skillet or baking dish.
To assemble, wedge the blackberries in between the bread slices making sure they are evenly dispersed. Next sprinkle the caramelized onions over the bread. Carefully pour the cream and egg mixture over everything. Finish by crumbling the goat cheese on top, pushing down into some of the cracks.
Let sit for 10 minutes before baking. Bake for 35-40 minutes or until golden brown and the middle is set.