Put the pine nuts, basil, mint, chives, garlic, lemon juice and the 1/4 cup olive oil in a tall bowl or blending container. Using an immersion blender, blend until smooth. Stir in the Parmigiano-Reggiano, and season the pesto with salt and pepper.
Preheat a panini press to 400F or medium-high heat.
Brush one side of each bread slice with olive oil. Lay the slices oiled side down and spread the top of each slice with 1 Tbs. of the pesto. Arrange 3 or 4 mozzarella slices in a single layer on 4 of the bread slices, then top with a layer of tomato slices, dividing evenly. Season with salt and pepper. Top each with one of the remaining bread slices, oiled side up.
Place 2 sandwiches on the preheated panini press, close the lid, and cook until the bread is toasted and the cheese is melted, 3 to 4 minutes. Transfer the sandwiches to a cutting board and cut in half. Repeat with the remaining sandwiches. Serve immediately.
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