Preheat oven to 375 degrees. Throughly rinse produce and pat dry. Peel and mince shallot. Stem dill fronds and coarsely chop. Rinse smoked salmon, pat dry, and cut into a 1/2" dice.

Melt butter in microwave or small pan and let cool slightly. Measure out 1/4 cup of flour and reserve. In a medium mixing bowl, whisk together remaining 1 cup flour, sugar, and a pinch of salt. In a second mixing bowl, combine melted butter, 2 Tbsp. liquid egg (reserve remaining for frittata), and 1/4 cup water. Add egg mixture to flour mixture and stir until dough comes together. Dough should be firm but not sticky. Add reserved flour if dough is sticky or add additional water, 1 Tbsp. at a time, if dough is too dry.

Spray underside of a rimmed baking sheet with non-stick cooking spray and place a damp towel underneath sheet. Place dough on prepared baking sheet and, with a rolling pin or wine bottle, roll dough as thinly as possible, 1/8" or thinner. Using underside of the baking sheet makes it easier to roll out, and helps dough stick to the sheet, helping the lavash to be as thin as possible.

Brush rolled out dough with 1 tsp. olive oil and season with za'atar spice and a pinch of salt and pepper. Bake for 15-20 minutes, or until dough is lightly browned and crispy. The texture should be paper-thin. Remove and let cool while frittata is being made. When cool enough to handle, break up into smaller pieces.

In a medium mixing bowl, combine remaining liquid egg with heavy cream, half the dill, half the feta, and half the diced smoked salmon (reserve remaining dill, feta, and salmon for garnish). Heat 1 tsp. olive oil in a small oven-safe non-stick pan over medium heat. Add minced shallot and cook for 2 minutes, or until translucent. Add egg mixture and cook, stirring gently, 2 minutes until eggs on bottom of pan begin to set. Move pan to oven and cook 15 minutes until eggs are fully set.

Serve frittata in pan or divide between two plates. Garnish with reserved diced smoked salmon, feta, and dill. Serve alongside lavash.