In a mixing bowl, combine the chopped marinated artichokes, chopped broccoli florets, Greek yogurt, shredded fontina or mozzarella cheese, fresh thyme leaves, chopped oregano, crushed red pepper flakes, and a pinch of kosher salt and black pepper. Mix until well combined, ensuring that all the ingredients are evenly distributed.
Lay out the slices of bread on a clean surface. Spread a generous layer of basil pesto on one side of each slice. On four slices, add a portion of the artichoke and broccoli filling, followed by slices of Brie. Top with the remaining bread slices, pesto side down, to create sandwiches.
Heat a skillet or grill pan over medium heat. Drizzle a little extra virgin olive oil to coat the bottom lightly.
Once the skillet is hot, carefully place the assembled panini in the pan. Press down gently with a spatula to ensure even toasting. Cook for about 3-4 minutes or until the bread is golden brown and crispy. Flip the sandwiches, adding more olive oil if necessary, and cook for another 3-4 minutes until the other side is equally golden and the cheese is melted.
While the panini are still warm, drizzle each sandwich with thyme honey. To make thyme honey, simply mix honey with fresh thyme leaves in a small bowl, allowing the flavors to meld.
Remove the panini from the skillet and let them cool for a minute. Cut in half and serve with extra thyme honey on the side for drizzling. Enjoy your Artichoke and Broccoli Pesto Brie Panini with Thyme Honey warm for the best experience!