1. Preheat a grill with a medium-high heat fire.
2. Gently mix together the chicken, sambal oelek, 11/2 teaspoons salt, and 3/4 teaspoon pepper in a large bowl. Form into 4 (4-inch) patties with an indentation in the center of each burger.
3. Brush the grill grate with 1 tablespoon oil. Lay the burgers on the grill and cook until grill marks appear, 5 to 6 minutes. Flip the burgers and continue grilling until an instant-read thermometer inserted sideways registers 160 degrees F, about 5 minutes. Set burgers aside to rest.
4. Lightly brush the pineapple rings with oil, then grill the rings turning once, until distinct grill marks appear, about 3 minutes per side.
5. Meanwhile, heat a large heavy nonstick skillet on the grill or on the stovetop over medium-high heat. Melt the butter and fry the eggs, sunny-side up, 3 to 4 minutes.
5. Tuck the burgers into the rolls, top with pineapple rings, beets, lettuce, tomato slices, and eggs. Serve.