Heat the oven to 200°F and a waffle iron to medium-high.
Whisk together the flour, baking powder, baking soda, kosher salt, and pepper in a large bowl. Stir in the parmesan, chives, and oregano until just combined.
In a separate medium bowl, whisk together the buttermilk, eggs, and butter. Pour the wet ingredients into the dry ingredients and gently fold with a spatula until just combined. Do not over mix. (Some lumps are OK and will result in a lighter waffle.) Set aside for 5 minutes.
Roughly chop the tomatoes and place them in a small bowl. Drizzle with olive oil and season with kosher salt and black pepper. In a second large bowl, lightly toss the arugula with olive oil and set aside for assembly.
Grease the waffle iron. Pour enough batter to fill the bottom of the iron without covering the grids into the iron and close the lid. Cook until golden brown and crisp, about 4 to 5 minutes. Store fresh waffles on a baking sheet in the warm oven. Repeat until all of the remaining batter is used.
Place the waffles on a serving board. Gently tear the burrata into pieces and position on top of each waffle, allowing the filling to spill out. Add the tomatoes, arugula, and torn basil leaves. Finish with hot honey and a pinch of flaky salt. Serve immediately.