Place oven rack in lower third of oven, and preheat to 400˚F. Brush the cut sides of the baguette with 2 tablespoons of olive oil. Arrange, cut sides up, on a baking sheet and bake until golden, about 10 minutes.
Heat a 2 1/2-quart saucepan over medium heat. Sauté the crushed garlic in the remaining tablespoon of olive oil for just one minute, just until fragrant. Pour the broth into the pot, then add the salt and pepper. Bring to a simmer, then stir in the frizen spinach and cheese. Simmer, covered, stirring occasionally, until spinach is just tender, about 8 minutes. Add the eggs in a slow stream while stirring constantly. Allow to simmer 30 seconds more, top with more freshly ground pepper, and serve with toasted baguette.