For pesto:Blanch the broccoli florets in salted water until crisp-tender.
Drain and immediately immerse the broccoli in ice-cold water to retain the bright green color.
Drain again, and transfer the broccoli to the work bowl of a food processor fitted with a metal blade.
Add parsley, garlic, pine nuts and Parmigiano.
Pulse 7 or 8 times until a chunky paste forms.
Pulse again while drizzling in the olive oil.
When the mixture reaches the desired consistency, stop and season to taste with salt.
Store pesto in an airtight container in the refrigerator for up to one week.
For cheese fingers:Spread 2 tablespoons of pesto on 1 slice of bread.
Top with a slice of cheese, then another piece of bread.
Repeat with pesto and remaining cheese and bread.
Preheat skillet over medium heat.
Melt 1 tablespoon of butter in skillet, and add a sandwich.
Place a weight, like a foil-wrapped brick, on the sandwich to press it down during the grilling.
When the one side is golden, add another tablespoon of butter, turn the sandwich over, and brown.
Repeat with remaining sandwiches.
Let the sandwiches cool for a bit and then trim the edges.
Cut the trimmed sandwiches into 4 fingers/rectangles.
The sandwiches may be made a few hours ahead and reheated before serving.