- Using center section of each potato, slice three horizontal 1-inch-thick slices (trim off ends of potatoes). Using a small round cookie cutter cut three circles (or use other cutter for desired shape). Reserve ends for other use. Soak cut potatoes briefly in cool water to prevent them from discoloring.
- Place potatoes in pot of heavily salted water; bring to simmer. Cook until potatoes begin to soften, 5 to 8 minutes; remove from water and dry at room temperature. Potatoes should be cooked 1/2 to 3/4 of the way through.
Prepare Bacon-Mustard Aïoli:
- In food processor, pulse yolks, lemon juice, zest and garlic until incorporated and smooth, about 1 minute.
- Combine oils; slowly add to egg mixture with processor running so mixture emulsifies. If mixture gets too thick, add 1 to 2 tablespoons milk or water. Transfer to small bowl.
- Using rubber spatula, fold in bacon, mustard and chives; season to taste with salt and pepper. Set aside in refrigerator for up to 2 days.
Prepare Cheddar Béchamel:
- Melt butter in small, heavy saucepan over medium heat. Stir in flour; cook to a smooth paste, about 1 minute.
- Whisk in milk; cook until mixture is thick and smooth, 2 minutes. Reduce heat to low, add cheese, salt and pepper; stir until cheese has melted, 2 minutes. Remove from heat; adjust seasoning. Reserve warm.
Prepare Arugula Cream:
- In a heavy saucepan, melt butter over medium-high heat. As butter begins to brown, add onion and garlic; cook until onion becomes translucent, 2 to 3 minutes.
- Add arugula, lemon zest, cream, salt and pepper. When cream reaches a simmer, lower heat; cook until liquid is reduced by half, 7 to 8 minutes, watching closely so cream doesn't boil over.
- Remove from heat; transfer to blender or food processor. Purée until smooth. Adjust seasoning. Reserve warm.
- Liberally season potatoes with salt and pepper. Heat oil in deep-fryer to 350°F to 360°F, and finish cooking potatoes for 6 to 8 minutes until fork tender and golden brown. Remove potatoes; carefully slide lollipop stick into each for service.
- Spoon 1 tablespoon each of aïoli, béchamel and arugula cream onto a plate. Carefully place potato lollipop so it stands secured in sauces. Garnish with minced chives. Serve at once.