Make crostini: Preheat oven to 375°. Slice baguette into ten 1/4-in.-thick slices on the diagonal (save leftover bread for another use). Brush both sides of each slice lightly with oil and sprinkle with salt and pepper.
Set slices on a rimmed baking sheet and bake until golden brown and crisp, rotating pan halfway through cooking, about 15 minutes total. Let cool.
Make topping: Whisk together 1/4 cup oil, vinegar, shallot, garlic, salt, and red pepper flakes in a small bowl.
Crack each quail egg into its own small bowl or cup (puncture the membrane beneath the shell with the tip of a knife). Heat a medium nonstick frying pan over low heat and brush with oil. Slide 5 eggs from bowls into pan, cover pan, and cook until the whites are set but the yolks are bright yellow, about 2 minutes. Transfer cooked eggs to a container lined with wax paper to bring to your picnic. Repeat with remaining 5 eggs.
Stir parsley and oregano into shallot mixture just before serving. Set a folded slice of beef on each crostini, drizzle with some sauce, and top with an egg.