Options for Removing Flat, Brown Facial Moles?

Surgical excision is probably the surest way to remove those moles but will leave a slight scar. These scars can be made to be less noticeable for you. Getting a biopsy, or pathological assessment is always something to think about. Lasers can help remove them without a scar but do not always completely remove them. Most of these are switched and have short pulses to target the moles without affecting structures around it and have less complications.
IPL is a newer possible treatment that can act like a laser but there is less experience with this approach. Lasers and IPL usually require more than one treatment and can require up to 4-6 treatments or more and the mole can come back sometimes.
Consulting a Board Certified Facial Plastic Surgeon would be something that I would recommend. We specialize in the face and are highly qualified to help you with this.

I do not have the luxury of seeing what these lesions look like. However, from your description you might have a condition called dermatosis papulosa nigra.
This is a relatively common condition principally in darker-skinned people. It consists of multiple, small pigmented papules, flat or filiform (projecting) on the face and neck. It tends to run in families. Bill Cosby, the American comedian, would be a good example of this condition.
Treatment is for cosmetic concerns only and consists of shaving the lesions off or using light electrodesiccation. Initially, only a few should be removed to be sure there is no hypo or hyperpigmentation.