I started sowing leeks and Zeebrune shallots in polystyrene tubs last year, which gives deeper rooting and continued growth for eight weeks to pencil size. JB Packaging sells a variety of sizes and depths.
As for sowing date, I have started sowing later to avoid the spring allium leaf miner breeding season: sowing after new moon in April is my go-to date.
Unless you want huge leeks, planting out mid June is perfectly timely and fits better with having an early crop first like radish, sprung turnip or broad beans.