Cut each tomato in half crosswise and grate on the coarse side of a box grater into a bowl, discarding the skins. (For canned tomatoes: crush by hand in the bowl, including all juices.)
Warm olive oil in a wide skillet over medium heat. If using garlic, cook 30 seconds until fragrant. Add the tomato pulp, season with salt, pepper, and sugar if needed. Stir to combine.
Cook over medium-high heat, stirring occasionally, for 10 to 15 minutes until the sauce thickens, deepens in color, and holds a trail when a spoon is dragged through it. Taste and season again.
Crack eggs into a bowl with a pinch of salt. Beat lightly with a fork -- just enough to combine yolks and whites.
Reduce heat to low. Wait 30 seconds. Pour the eggs over the sauce. Let sit 15 to 20 seconds, then fold gently with a wooden spoon in slow, sweeping strokes until soft curds form throughout the sauce.
When eggs are about 80% set -- still slightly underdone and glossy in places -- remove the pan from the heat. Residual heat finishes them.
Spoon into bowls. Crumble feta generously on top. Add fresh oregano and a drizzle of olive oil. Serve immediately with crusty bread.