In a medium skillet, heat oil over medium-high heat. Add the garlic, red pepper flakes, and oregano; cook, stirring, until fragrant, about 1 minute. Raise heat to high; crush each tomato when adding to pan. Season with salt and pepper. Cook the tomatoes, breaking them up with a spoon, until thickened slightly, 2 to 3 minutes. Stir in the basil.
Lower the heat to medium; stir in 2 large egg whites (reserve the remaining egg whites for another use). Cook the egg whites, stirring, until set and the tomato sauce thickens, about 1 minute.
Remove the skillet from the heat. Using the back of a spoon, make 4 indentations in the sauce; place an egg yolk into each indentation. Draw the sauce in from edges of the pan around each yolk and cover the skillet. Set aside until the yolks are warmed through and beginning to set, about 3 minutes. To serve, spoon the sauce with a yolk onto a piece of toast.
This recipe was provided by professional chefs and has been scaled down from a bulk recipe provided by a restaurant. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.