In a small saucepan over medium heat, bring the mustard and 1/4 cup water to a simmer. Whisk in the goat cheese until combined. Set aside; the mixture will thicken as it cools.
Toast the mini bagels until golden brown and crispy. Set aside.
In a medium nonstick skillet, heat 1 teaspoon of the olive oil over medium heat. Add 4 slices of the pancetta and cook until golden brown and crispy on both sides, 2 to 3 minutes per side. Drain the pancetta on paper towels. Repeat with the remaining pancetta.
Heat the remaining 1 teaspoon olive oil in a small frying pan over medium-low heat. Crack 4 quail eggs into the pan, sprinkle with salt and pepper, and cook until the whites are set but the yolks are still runny, 2 to 3 minutes. Remove to a plate and repeat with the remaining eggs.
To build the PLT: Spread each bagel half with 1 tablespoon of the goat cheese sauce. Layer with a slice of pancetta, a slice of tomato, and some arugula. Top the whole thing with a fried quail egg and a grind of freshly ground black pepper. Serve.
Recipe courtesy of Giada De Laurentiis