Pour enough vegetable oil into a small saucepan or deep skillet to a depth of about 1/2–3/4 inch and heat to 350°F (175°C).
Add the sliced shallots and fry, stirring often, until golden brown and crisp, about 6–8 minutes; transfer with a slotted spoon to paper towels to drain.
Fry the sliced garlic in the hot oil until just barely golden, about 1 minute; transfer to paper towels to drain and set aside with the shallots.
Stir the pressed garlic into the sesame oil in a small bowl and set aside.
Bring the chicken broth to a boil in a large saucepan.
Pulse the cubed chicken in a food processor until coarsely ground, then add it to the simmering broth and cook, stirring to break up clumps, until cooked through, about 5 minutes; use a slotted spoon to transfer the cooked chicken to a large bowl.
Toss the cooked chicken with 1 tablespoon of the fish sauce in the bowl and set aside.
Return the broth to a boil, add the rice stick noodles, and cook until tender, 3–4 minutes; lift the noodles out with a spider or slotted spoon and add them to the bowl with the chicken.
Bring the broth back to a boil, add the beans (or asparagus) and cook until crisp-tender, about 5 minutes; transfer them to the bowl with the chicken and noodles.
Add the sesame-garlic oil, lime juice, 1 tablespoon sugar, cayenne, and the remaining 3 tablespoons fish sauce to the bowl with chicken, noodles, and vegetables; toss well to coat everything evenly.
Transfer the noodle mixture to a large shallow serving bowl and sprinkle with cilantro, green onions, chopped peanuts, fried shallots, and fried garlic; sprinkle the remaining sugar over the top if desired.
Serve immediately with hot chili sauce (such as sriracha) on the side.
