In a large skillet, bring 1/4 inch of water to a boil. Add the spinach and cook, stirring, until just wilted, about 1 minute; drain and press out as much water as possible. Wipe out the skillet.

In the same skillet, heat the 2 tablespoons of olive oil until shimmering. Add the garlic and anchovy paste and cook over high heat, stirring, until fragrant, 1 minute. Add the spinach, season with salt and stir just until coated, about 10 seconds.

Light a grill or preheat a grill pan. Using slightly moistened hands, form the sausage meat into four 4-inch patties, about 3/4 inch thick. Brush the burgers with oil and grill over moderate heat until browned and crusty on the bottom, about 5 minutes. Carefully flip the burgers. Top with the cheese and grill until the burgers are cooked through and the cheese is melted, about 5 minutes longer. Spread the pesto on the rolls. Top with the burgers and spinach and serve.