Your progress on the Dawning quests is shared account-wide, so it doesn't matter which character you use. As long as you've picked up the quests on all your characters, they will all reflect your most current progress. For "Share The Dawning Spirit," as indicated by the "As a community..." part in the quest description, this is a community wide-quest. You have to make and deliver the specified treat yourself, but the part for spreading Dawning Spirit is progress by the entire Destiny 2 community. As you deliver treats, turn in Dawning bounties, and claim Dawning triumphs, you will see that you're earning Dawning Spirit. Every time that happens, it contributes towards the total percent completed for the quest, and the Dawning Spirit earned by every other player contributes as well. The reason it may appear that you're "stuck," is that the required amount of Dawning Spirit increases for each step in the quest, with the final goal to complete the entire thing being 1 Billion Dawning Spirit, so it may go up faster or slower at different times, as the number of players goes up and down throughout the day. Once you complete the "And To All A Good Flight," quest, you get an upgrade for the Dawning exotic ship, which if you have the ship equipped, gives you a chance to earn Dawning Spirit every time you earn an ingredient for the oven, which should help speed up progress, as more people complete that quest.