In a heavy bottomed pot, heat the 3 tablespoons of the thick coconut cream to high for about 1 minute. When the cream starts to sizzle, stir in the curry paste into the cream like building a roux. Add the onion and half of the makrut lime leaf chiffonade. Stir-fry the paste until the paste starts to thicken, dry out, and become fragrant, about 1 minute. If the curry starts to sputter, add a small amount of coconut milk to keep the paste moving. Cook the paste until it has the consistency of peanut butter.
Stir in the remaining coconut milk into the curry paste. Increase the heat to high until you reach a full rolling boil. Allow the curry to boil until it reduces by about 25% or coats the back of a wooden spoon, 10 to 15 minutes.
Reduce the heat to a simmer. Add the chicken, tamarind paste, fish sauce and sugar. When the chicken is about halfway cooked, about 5 minutes, add the bell pepper and zucchini. Let simmer until the chicken is cooked through, about 5 minutes. Add the bamboo shoots and Thai basil leaves. Taste and adjust seasoning. Serve immediately over jasmine rice and garnish with a few ribbons of the remaining makrut lime leaf.
Never shake the can of coconut milk! You want the cream to naturally separate and rise to top. You can use that natural coconut cream like cooking oil to fry your curry pastes!