Preheat oven to 375F.
In a saucepot over low heat, add oil, butter, garlic and a pinch of salt. Cook on low until butter melts and the pot is fragrant with the scent of garlic, about 5 minutes. Turn off heat and set aside.
In a saut pan over medium heat, add half of the reserved garlic oil, onions, broccoli and paprika. Stir while cooking until onions are tender but broccoli is still bright green with a bit of crunch left -- go ahead, give it a taste! Season with a bit more salt if needed. Allow to cool.
With a serrated knife, cut about 2 inches off one tip of each baguette; reserve the trimmed ends. Using the knife, hollow out the inside of each baguette leaving just an 1/8 inch of bread around the edges. Be careful to not puncture the outside of the bread from the inside while hollowing each baguette. Save the scooped-out bread for breadcrumbs.
In a large bowl, add the cooled vegetables, cheddar cheese and mozzarella. Stir to combine.
Use a spoon to fill each baguette with the filling. Use the handle of a wooden spoon or something similar to pack the stuffing in tight. When done with each, invert the reserved trimmed end of each baguette into the open end of each baguette to create a seal. Tightly wrap each in aluminum foil and bake for 25 minutes. Remove from the oven and raise the temperature to 400F. Open the aluminum foil and brush both with remaining infused oil. Return to the oven uncovered and cook until golden, about 5-8 minutes more.
Allow baguettes to cool slightly and, using a serrated knife, cut 1-inch slices along the length of each baguette. If transporting the baguettes to a party, slice and keep them in the form of a baguette, then wrap them back up in the same aluminum foil. These are great cold or room temperature -- feel free to slice, cool completely then re-wrap and refrigerate.
Preparation time: 10 minutes
Cooking time: 40 minutes
*If using a regulation-sized baguettes instead of demi baguettes, cut them in half or thirds to make it easy to hollow out. Leave ends intact and in the absence of trimmed-ends to plug the fillings, just tightly wrap each baguette section. Also, whether using demi or regulation-sized baguettes, pick out ones that are more rounded than flat, ahhhthe easier to stuff you with.