Toss lemon slices and garlic in sugar. Set aside.
Slice chicken breasts in half, place between two pieces of plastic, and pound to about ¾ inches thick (no need to be precise, just make sure all of the chicken pieces are about even thickness). Season chicken pieces on both sides with salt and black pepper.
Place ½ cup flour in a shallow baking dish.
Heat the largest skillet you have over medium heat. Add 1 tablespoon olive oil and swirl to coat the bottom of the pan. Working quickly, dredge each piece of chicken in flour, shake off the excess, and place the chicken in the hot pan (if it won't all fit in a single layer, cook the chicken in batches). Cook on both sides until deep golden brown and cooked through, 5 to 7 minutes total. Set chicken aside and return pan to heat.
Add 1 tablespoon olive oil to heated pan and then lemon slices and garlic. Saute until garlic is fragrant and lemons are starting to turn golden in spots, 1 to 2 minutes (watch it closely and remove it from heat if the garlic starts to turn brown). Pour charred lemons and garlic back into the bowl they were marinating in and return pan to heat.
Add shallots and oregano to heated pan. Cook until shallots are tender, ~1 minute. Add 1 tablespoon flour and stir until no dry spots of flour remain. Add white wine and stock to pan, scraping up any browned bits on the bottom of the pan. Increase heat to bring sauce to a simmer. Simmer until sauce reduce by half, about 5 minutes. Remove pan from heat and whisk in butter and capers until butter melts.
Return chicken pieces to pan and spoon sauce over top. Pour lemons and garlic over top and sprinkle with parsley.
Serve over spaghetti or rice.