Heat broiler, with rack set 4 inches from heat. Place bread on a baking sheet, and brush both sides with 2 tablespoons oil. Season with salt and pepper. Broil until golden, 1 to 3 minutes per side; set aside.
In a large nonstick skillet, heat 1 teaspoon oil over medium. Add scallions and as much spinach as will fit; season with salt and pepper. Cook until wilted, tossing and adding more spinach as room becomes available, 2 to 3 minutes. Drain off excess liquid; mix in goat cheese. Transfer to a bowl; cover to keep warm. Set aside.
Wipe out skillet; heat remaining 2 teaspoons oil over medium. Gently crack eggs into skillet; season with salt and pepper. Cook until whites are almost set, about 1 minute. Cover, and remove from heat; let stand until whites are set but yolks are still soft, about 3 minutes.
Top each piece of toast with spinach mixture and 1 egg; serve immediately.
Baby arugula doesn't last long, so buy only what you plan to use within a few days. To clean, fill a bowl with cold water, and add greens, swishing gently. Scoop out greens. Drain water off. Repeat with fresh water until no grit remains at the bottom of the bowl. Dry with a salad spinner, or pat with paper towels. Store washed greens in a resealable plastic bag lined with dry paper towels in the refrigerator up to 1 day.