You could say that it is Pellau (west indian spelling

), there are so many different names for a dish that is prepared everywhere with slight variations.
I made a version of this the night I posted the recipe. I dont have a physical-written-on-paper recipe for mine but it sounds very close to yours. Unfortunately, Im one of those who cooks by instinct,

, so I don't have accurately measured ingredients. I dont think any west indian has actual accurately measured ingredients. Here is the gist of the one I made:
1 large onion chopped (I love onion but you can use a smaller one)
2 cans chicken broth
2 cloves garlic, smashed
Couple pinches of thyme
Salt & Pepper to taste
2 shakes of red pepper flakes
bone in chicken pieces, cut into pieces (the bones are added too)
Cajun sausage, sliced then chopped
a little veggie/olive oil to saute
a pat of unsalted butter
Saute onion, garlic, thyme, and red pepper flakes. Add chicken and sausage and brown a little. Remove. Add rice and cook, stirring, until traslucent and coated. Return chicken and sausage to pan. Add broth bring to a boil then lower to a simmer. Add pat of butter and cover. Simmer till meat is done and rice is cooked. Adjust seasoning and serve hot with pickled cucumber and lite salad. Mainly just a few lettuce leaves and tomato.
Pickled Cucumbers
1 or 2 cucumbers peeled, scored lengthwise with a fork and sliced thin
Salt & Pepper
Half a small onion chopped fine
Juice of 1 lime
A few drops of Hot Sauce
Few thyme leaves
In a small bowl combine all ingredients, adjust seasonings and refrigerate until ready to serve.
I call it pickled cucumbers. Im not sure what other Bajans call it but I had to name it something.

Its a staple during Sunday dinner along with Coleslaw, Potatoe Salad, Plain Salad with no dressing and Soda mixed with Coconut water. Dessert for kids is when the ice cream man comes around.

Fond memories.
Edit: I forgot to mention. Before cooking my chicken I do what all Bajans do. Clean the chicken, trim off any excess fat and the skin, sprinkle with salt and squeeze the juice of 2 limes over it. Let it sit in the fridge for about 10 minutes or until you are finished prepping everything else. Its usually the latter one. When ready to cook.....rinse chicken, cut slits near the bone and stuff with Bajan seasoning. Continue with recipe. For this recipe you can just coat chicken with a thin layer of seasoning. I was told that "lime and salting" removes the raw taste. My mom has another explanation that Im not supposed to share with anyone
Im a little embarrassed about my recipe writing skills. I hope these are understandable.