Whisk together tempura flour, egg, and water until smooth with no lumps. Let the batter rest for 10 minutes.
Season the batter with salt, pepper, cayenne pepper, and vermouth to taste. Whisk again.
Wash the chicken breast fillets and pat completely dry. Cut into strips approximately 10 cm long.
Heat palm fat in a pan to at least 1 cm depth over medium-high heat until simmering.
Lightly coat each chicken strip with flour. Dip completely in the tempura batter.
Place each coated strip into the hot fat. Fry for 5–6 minutes, turning 2–3 times, until golden brown all over and the internal temperature reaches 74°C (165°F).
Remove strips with a slotted spoon and drain on paper towels.
Serve immediately. If other ingredients are not ready, keep warm in an oven at 70°C (158°F).
Serve with Asian-style basmati rice and wok vegetables, or Italian-style with spicy tomato sauce on tagliatelle and parmesan.