Position a rack in the center of the oven and heat the oven to 450°F.
On a small rimmed baking sheet, toss the bread with 2 Tbs. of the olive oil, 1/4 tsp. salt, and 1/4 tsp. pepper. Spread in a single layer and bake until golden, about 8 minutes.
Meanwhile, heat the remaining 2 Tbs. oil in a 4-quart saucepan over medium-high heat. Add the onion and cook, stirring occasionally, until it begins to brown, about 4 minutes. Add the garlic and cook, stirring, until fragrant, about 30 seconds. Add the escarole and cook, stirring, until tender, about 5 minutes. Add the chicken broth, 1 tsp. salt, and 1/2 tsp. pepper, bring to a boil, and remove from the heat. Whisk the eggs with the lemon juice and then whisk the egg mixture into the hot soup.
Divide among 4 bowls and then top with the croutons, small dollops of the pesto, and the sun-dried tomato, if using.
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