Preheat the oven to 425 degrees F and place a rack on the second-to-top position. Line a rimmed baking sheet with foil leaving a 2-inch overhang on each side. Set aside.
Whisk together 1 tablespoon of the olive oil, the honey and half of the lemon juice and zest in a mixing bowl. Cut the cucumber in half and dice one half into 1/4-inch pieces. Add the cucumber to the bowl with the vinaigrette along with the grape tomatoes. Toss to combine and season with salt and pepper. Cover with plastic wrap and refrigerate.
Grate the other half of the cucumber on a box grater over a mixing bowl. Wrap the grated cucumber in a kitchen towel and squeeze out any excess liquid over the bowl. Reserve the liquid. Place the grated cucumber into another bowl and add the remaining lemon zest and juice, the remaining 1 tablespoon extra-virgin olive oil, the yogurt, dill and parsley. Stir to combine and season with salt and pepper. Cover with plastic wrap and refrigerate.
Add the hummus to the bowl with the reserved cucumber liquid and whisk to combine. The hummus should be thin enough to drizzle; if not, add 1 tablespoon water at a time until you reach the proper consistency.
Spread the pita chips onto the prepared baking sheet in a single layer, allowing some overlap. Scatter the kasseri cheese evenly on top and bake until the cheese is melted and the pita chips are warmed through, 2 to 3 minutes.
To serve, scatter the shredded chicken evenly all over the pita chips. Drizzle with the hummus and top with the marinated cucumbers and tomatoes using a slotted spoon. Garnish with dollops of the cucumber yogurt sauce and kalamata olives.