In a medium-size saucepan over medium-high heat, add 1 tablespoon olive oil and cook lamb, breaking up the meat with a wooden spoon. With a slotted spoon, remove lamb to a plate to cool. Wipe the pan clean.
Over medium heat, add 1 tablespoon olive oil to pan and cook onion and garlic gently until softened and translucent. Add cumin and oregano and stir well. Cook just 1 minute to release fragrance. Remove from heat and let cool. Add lamb and mix well.
In a medium-size bowl, combine arugula, remaining olive oil, and lemon juice. Season to taste with salt and pepper.
Heat your grill to medium-high, and oil the grates lightly. Roll dough out as thinly as possible. Place dough on grates and cook about 1 minute, or until very lightly browned. Turn dough over and quickly sprinkle with lamb mixture and feta. Cover the grill with the lid. Let cook about 2 minutes and check; edges should be browned. Spoon arugula mixture over the top.
In a standing mixer with dough hook, combine whole-wheat flour, all-purpose flour, yeast, salt, and sugar. With the motor running, gradually add oil and water. Knead on low speed until dough is firm and smooth, about 10 minutes.
Divide dough into four balls. Line two cookie sheets with parchment paper. Place two balls on a sheet and cover with a damp towel. Let them rise in a warm spot until they have doubled in bulk, about 2 hours.