**Core Concept**
The patient's symptoms suggest a possible orbital or adnexal pathology, specifically a condition that causes proptosis (bulging of the eye). Proptosis can be due to various causes, including thyroid orbitopathy, orbital tumors, or vascular anomalies. The absence of redness or pain makes an inflammatory cause less likely.

**Why the Correct Answer is Right**
In this scenario, the investigation of choice is Orbital Imaging, typically using MRI or CT scans. These imaging modalities can help differentiate between various causes of proptosis, such as thyroid orbitopathy (e.g., Graves' disease) or orbital tumors (e.g., lymphoma, metastasis). MRI is particularly useful for assessing soft tissue involvement, while CT scans provide better detail of bony structures. The choice of imaging modality depends on the suspected diagnosis and the patient's overall clinical presentation.

**Why Each Wrong Option is Incorrect**
**Option A:** Orbital Ultrasound is not the first-line investigation for unilateral proptosis, as it may not provide sufficient information about the extent of disease or the involvement of surrounding structures.

**Option B:** Orbital Biopsy is an invasive procedure and should only be considered when imaging results are inconclusive or when a specific diagnosis is suspected based on clinical and imaging findings.

**Option C:** Thyroid Function Tests (TFTs) may be helpful in evaluating patients with suspected thyroid orbitopathy, but they are not the primary investigation for unilateral proptosis.

**Clinical Pearl / High-Yield Fact**
In cases of proptosis, it's essential to consider both the ophthalmic and systemic implications. A thorough examination and imaging studies can help identify potential causes, which may require multidisciplinary management.

**Correct Answer:** C. CT Scan of the orbits.