Make the dressing. In a mason jar, combine the olive oil, red wine vinegar, lemon juice, sumac, oregano, cumin, honey, and dijon. Use an immersion blender to blend the vinaigrette until smooth and then season to taste with salt and pepper. Alternately, whisk all the ingredients together by hand until emulsified.
Make the pita chips. Line a baking sheet with parchment paper. Open the pita pockets and tear the pita into pieces. Toss with the olive oil and sprinkle with the za'atar, sumac, and salt. Bake at 375 F for 6-8 minutes, until golden brown and crispy. Remove from the oven and let the pita cool.
Prep the lettuce. Tear the lettuce leaves off the head of lettuce and into bite-sized pieces. Rinse them in cold water and pat them dry or use a salad spinner to dry the lettuce. Lay the lettuce on a baking sheet and refrigerate until ready to use.
Marinate the vegetables. In a large mixing bowl, add the chopped tomatoes, cucumbers, sliced onion, and radishes, and half of the torn herbs and toss them with a few spoonfuls of the dressing and a sprinkle of salt and pepper. Let them marinate for 15 minutes.
Assemble the fattoush salad. Toss the marinated vegetables with the lettuce and pita crisps with more dressing right before serving. Top with the remaining fresh herbs and serve right away.