Combine 1 cup of flour, 1 teaspoon sugar, salt and yeast in the bowl of an electric mixer. Gently mix to incorporate. Heat the water and two tablespoons of the oil until warm. Add the water, oil and egg to the yeast mixture. Blend until moist, about 2-3 minutes. Stir in 1 3/4 cup flour while beating, and continue to beat until a dough forms.
Working on a floured surface, knead the remaining flour into the dough. Form the dough into a ball and cover with a bowl for 5 minutes. Meanwhile, generously grease a baking sheet with about 2 tablespoons of oil (it should look like too much oil). Press the dough and flatten it across the sheet (it should cover about half of a full sized baking sheet). Cover with plastic wrap and a dry towel and place in a cool oven for 30 minutes.
In a medium skillet, cook the rosemary, cherries, lemon zest and 1-2 tablespoons of oil for about 5 minutes (just enough that the cherries begin to soften). Remove from heat and allow to cool.
Preheat the oven to 200 degrees. Uncover the dough. Grease the dough with about 1 tablespoon of oil. Turn the dough over and grease the other side with about 1 more tablespoon of oil. Create indents in the dough with a wooden spoon handle (at approximately 1-inch intervals). Spoon the cherry mixture across the dough, leaving a 1-inch border. Bake the focaccia until it is golden and the edges are crisp, about 20-25 minutes. Remove from the baking sheet and slide onto a wooden cutting board. Slice into generous pieces and enjoy.