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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.
Wine Pairing: Italian sausage needs a robust red wine like Zinfandel. Coincidentally, superstar Zinfandel producer Peter Seghesio and his family make their own Italian sausage every year to give as holiday gifts. Look for his basic 2008 Seghesio Sonoma Zinfandel or the single-vineyard 2008 Seghesio Home Ranch bottling.