**Core Concept**
The patient's presentation of a persistent cough and positive sputum culture for a broad budding yeast on Sabouraud Dextrose Agar (SDA) suggests an opportunistic fungal infection, likely caused by a pathogen that takes advantage of the patient's immunocompromised state due to HIV infection.

**Why the Correct Answer is Right**
The presence of broad budding yeast cells in the sputum culture is characteristic of *Pneumocystis jirovecii* pneumonia (PCP), a common opportunistic infection in HIV-positive individuals. *P. jirovecii* is a fungus that primarily infects the lungs and can cause severe pneumonia in immunocompromised patients. The infection occurs when the patient's CD4 cell count falls below 200 cells/μL, making them susceptible to opportunistic infections.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because *Histoplasma capsulatum* typically presents with a more subacute course and is often associated with exposure to bird or bat droppings, not a broad budding yeast morphology.
**Option B:** This option is incorrect because *Candida albicans* typically presents as a commensal organism in the mouth or vagina and can cause mucocutaneous infections, not a broad budding yeast morphology in the sputum.
**Option C:** This option is incorrect because *Aspergillus fumigatus* typically presents with a more acute course and is often associated with a history of asthma or other underlying lung disease, not a broad budding yeast morphology.

**Clinical Pearl / High-Yield Fact**
In HIV-positive patients, a persistent cough and positive sputum culture for a broad budding yeast should raise suspicion for PCP, which requires prompt treatment with trimethoprim-sulfamethoxazole (TMP-SMX) or alternative therapies in patients with a sulfa allergy.

**Correct Answer:** D.