Grilled sunflowers?

We grow “Teddy Bear” sunflowers, which are dwarf variety, and with compact “pom-pom” like flowers. Picked the “slightly wilting” blooms. Removed petals. Brushed head with oil. Grilled 10-15(?) min. Maybe we didn’t grill long enough? But the texture was…, uh, not pleasant. All stringy. There wasn’t really much to eat, even if it had tasted better. Maybe I need a larger variety of sunflower.