For an empowered Applied Researcher, does her discount apply to flow costs? I want to play Dredge Up, do I pay 1 or 2?

Yes — you pay 1 Energy, not 2.

Applied Researchers' Empowered ability ("Your spells cost [1] [A] less, to a minimum of [1]") is a discount on the total cost of your spells. Flow replaces Dredge Up's base cost with its Flow cost of [2] during finalization (Rule 829.1.c.1), and total-cost discounts are then applied to that result (Rule 356.4). So 2 − 1 = 1, and the "minimum of [1]" is respected — Dredge Up can't drop to 0 this way.

Same logic as her interaction with Deflect (FAQ #11833): the discount hits the final total cost, not just the printed number in the corner. Just make sure she's actually Empowered when you finalize the spell.