In a 10-inch frying pan over medium heat, stir half the olive oil and the 2 cloves garlic until fragrant, being careful not to brown garlic, about 1 minute. Pour the oil and garlic into a food processor. Wipe out the pan and set aside.
Add white beans, lemon juice, and salt to the food processor and whirl until smooth. Pour into a serving bowl.
Return the frying pan to medium heat and add the remaining 2 tablespoons olive oil and the rosemary sprigs. Warm the rosemary in the olive oil until fragrant, about 3 minutes, stirring occasionally so the rosemary doesn't burn. Remove from heat and let cool 10 minutes.
Set the rosemary aside and drizzle the olive oil over the bean dip. Mince one teaspoon of the rosemary leaves and sprinkle over the dip.
Party short-cut: Use the white bean dip as the foundation for an abundant tray of store-bought snacks, including hummus and baba ghanoush, olive tapenade, carrot sticks, and other vegetables. Serve with breadsticks, pita chips, and a thinly sliced baguette.
Do-ahead tips: Make dip up to 2 days ahead. Store dip in an airtight container in the refrigerator. Bring dip to room temperature before serving. Prepare the rosemary oil right before serving.