TO MAKE THE SHALLOT CRISPS Set a plate lined with a double thickness of paper towels next to the stove.
Heat the vegetable oil in a 6-inch saute pan over medium-high heat until hot but not smoking. (Test the temperature of the oil by adding 1 slice of shallot to the hot oil.
If the oil begins to bubble and sizzle without splattering, and the shallot turns golden, then the oil is ready.) Using a slotted spoon, add half of the shallots to the oil.
Fry until they are crisp and turn a dark golden brown, about 1 to 2 minutes.
Remove from the oil and drain on paper towels.
Repeat with the remaining shallots.
Set aside at room temperature until ready to garnish the beans.
The shallot crisps can be made several hours ahead.
TO SAUTE THE BEANS Heat a 12-inch saute pan over medium-high heat.
Add the butter and olive oil, swirl to coat the pan, and then add the green beans.
Saute, stirring frequently, until the beans are bright green and crisp-tender, about 5 minutes.
Season with salt and pepper to taste.
Transfer the beans to a warmed serving bowl and garnish with the shallot crisps.